Paper Title: Infinitesimal 2-braidings from 2-shifted Poisson structures.




Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

    Lise-Marie Imbert-GérardDepartment of Mathematics, University of Arizona, USA  (mailto:lmig@arizona.edulmig@arizona.edu) ,
Andrea MoiolaDepartment of Mathematics, University of Pavia, Italy       (mailto:andrea.moiola@unipv.itandrea.moiola@unipv.it) ,
Chiara PerinatiDepartment of Mathematics, University of Pavia, Italy (mailto:chiara.perinati01@universitadipavia.itchiara.perinati01@universitadipavia.it) , 
Paul StockerFaculty of Mathematics, University of Vienna, Austria (mailto:paul.stocker@univie.ac.atpaul.stocker@univie.ac.at)
    August 5, 2024
===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
It is shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra A defines a very explicit infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finitely generated semi-free A-dg-modules. This provides a concrete realization, to first order in the deformation parameter ħ, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.






  
*Keywords: derived algebraic geometry, shifted Poisson structures, L_∞-algebras, quantization, braided monoidal 2-categories





  
*MSC 2020: 14A30, 17B37, 18N10, 53D55












§ INTRODUCTION AND SUMMARY


Quantum groups are interesting and important mathematical objects
which enjoy a wide range of applications in representation
theory, topological quantum field theory
and integrability, see e.g. <cit.> for surveys.
There exist two complementary main approaches to the study of quantum groups:
The algebraic approach based on quasi-triangular Hopf algebras
and the categorical approach studying linear braided monoidal categories.
The relationship between these approaches is given by the observation 
that a quasi-triangular Hopf algebra has a linear braided monoidal category
of representations from which it can be recovered by Tannakian reconstruction.
An important class of examples of quantum groups are the quantized 
universal enveloping algebras U_q() which are defined
as quasi-triangular Hopf algebra deformations of the enveloping algebra U()
of a Lie algebra . In the case of formal deformations q=e^ħ, 
their linear braided monoidal representation categories 𝐑𝐞𝐩_ħ() can also be obtained 
directly as braided monoidal deformations of the linear symmetric monoidal representation
category 𝐑𝐞𝐩() of the Lie algebra , see
e.g. <cit.> and <cit.>.


Given the importance of quantum groups, it is justified and natural to search 
for a generalization of such structures to higher category theory.
Such `higher quantum groups' are then expected to have applications
in higher-categorical representation theory, higher-dimensional
topological quantum field theory and higher-dimensional integrability.
There have been various attempts to develop a theory
of `higher quantum groups' based on `higher Lie algebras' such as Lie 2-algebras, 
reaching from algebraic approaches,
see e.g. <cit.>, to more categorical ones,
see e.g. <cit.>. These attempts are distinct from 
the approach to `higher quantum groups' via `categorification' in the sense 
of <cit.> where 
`higher quantum groups' are constructed based on classical Lie algebras.


In the present paper, we initiate a new approach to `higher quantum groups'
which is rooted in recent developments in derived algebraic geometry,
in particular the theory of shifted symplectic and shifted Poisson structures on 
derived stacks <cit.>. Our inspiration 
comes from Safronov's results in <cit.> which show that the semi-classical
data associated with quantum groups, namely Lie bialgebra structures
and invariant symmetric tensors, arise naturally as 1 and 2-shifted Poisson structures
on the classifying stack BG=[pt/G] of an algebraic group G,
or its infinitesimal analogue given by the formal classifying stack 
B=[pt/] of the Lie algebra  of G.


Given an n-shifted Poisson structure on a derived stack X, for an integer 
n≥ 1, the symmetric monoidal dg-category  𝐏𝐞𝐫𝐟(X) 
of perfect modules over X can be quantized using the 𝔼_n-monoidal 
deformation quantization results from <cit.>. 
In the case where X = B, one has an equivalence
of symmetric monoidal dg-categories
𝐏𝐞𝐫𝐟(B) ≃𝐝𝐠𝐑𝐞𝐩()
identifying perfect modules over B with (homotopy-coherent) 
representations of  on cochain complexes. Hence, every 
2-shifted Poisson structure on B, which by <cit.>
is an invariant symmetric tensor in (^2)^,
defines a braided monoidal deformation 𝐏𝐞𝐫𝐟_ħ (B)
≃𝐝𝐠𝐑𝐞𝐩_ħ() which is expected to generalize
the constructions in <cit.> and <cit.> 
to representations of  on cochain complexes. In Corollary <ref>
we provide a concrete realization of this braided monoidal deformation to all orders of ħ.


Our working hypothesis is that the above picture should generalize
to the case where the Lie algebra  is replaced by a `higher Lie algebra', 
i.e. an L_∞-algebra L. Inspired by <cit.>, 
we will define the semi-classical `higher quantum group' data
on L to be given by the 2-shifted Poisson structures
on the formal classifying stack BL = [pt/L],
and propose the braided monoidal deformation 𝐏𝐞𝐫𝐟_ħ (BL)
≃𝐝𝐠𝐑𝐞𝐩_ħ(L) from <cit.> as a model for
the representation category of the `higher quantum group U_q(L)' with q=e^ħ.



It is worthwhile to emphasize that the existence results for deformations in <cit.> 
are quite implicit as they rely on formality of the 𝔼_n+1-operads for n≥ 1, 
see also <cit.>. Hence, it is a priori not evident how these abstract methods
relate to the more explicit deformation techniques for ordinary Lie algebras 
from <cit.> and <cit.>, which are based on the simple but powerful concept 
of an infinitesimal braiding, controlling the first-order deformation in ħ,
and its lift to a full deformation via a choice of Drinfeld associator.
The main goal of this paper is to display an explicit relationship between the 
abstract deformation results of <cit.> and infinitesimal braidings. This is done 
by adopting a more concrete and computational approach to derived algebraic geometry
which is rooted in Pridham's works <cit.>.


We will now explain our results by outlining the content of this paper.
In Section <ref>, we recall some basic aspects of derived algebraic geometry
that are needed in our work. In particular, we provide a brief discussion of the crucial concept of an 
n-shifted Poisson structure (Definition <ref>)
on a finitely generated semi-free commutative differential graded algebra 
(CDGA) A. By restricting this discussion to the stacky-affine case, 
the treatment remains self-contained in the language of CDGAs and their dg-modules. 



In Section <ref>, we study deformations of the symmetric monoidal dg-category 
_A^fgsf of finitely generated semi-free
A-dg-modules into a braided monoidal category.
More precisely, in order to have good control on 
the tower of homotopy coherences which can enter such deformations,
we restrict our attention to the simpler problem of deforming the homotopy 2-category

    _A := 𝖧𝗈_2(_A^fgsf)

of this symmetric monoidal dg-category.
The homotopy 2-category _A has the same underlying objects, 
i.e. finitely generated semi-free A-dg-modules, and its morphisms
are given by the good truncations to cohomological degrees {-1,0} of the 
morphism complexes in _A^fgsf, see Appendix <ref>.
We show in Proposition <ref> that, 
to first order in the deformation parameter ħ, such deformations
are governed by infinitesimal 2-braidings as in Definition <ref>. 
Our concept of infinitesimal 2-braidings is compatible with the one proposed
by Cirio and Faria Martins <cit.> in the context of categories enriched over 2-vector spaces.


Our main results are presented in Subsection <ref>,
where we show that every 2-shifted Poisson structure on a finitely generated semi-free CDGA
A defines, by a very explicit construction, a γ-equivariant 
infinitesimal 2-braiding on _A, see in particular Theorem <ref>. This means that
every 2-shifted Poisson structure on A defines an explicit first-order
braided monoidal deformation of the homotopy 2-category of the symmetric monoidal
dg-category _A^fgsf, which realizes concretely 
the abstract deformation results from <cit.> to first order in ħ.


In Subsection <ref>, we explore the behavior
of our first-order deformations at higher orders in the deformation parameter ħ.
Following the conventional theory of infinitesimal braidings from
<cit.> and <cit.>,
we observe in Proposition <ref> that
a γ-equivariant infinitesimal 2-braiding defines 
an all-order in ħ braided monoidal deformation of the homotopy 2-category _A 
provided that certain identities which are similar to Drinfeld associators hold true strictly.
We observe in Proposition <ref> that, in general, these identities
do not hold true strictly, but only up to homotopies which we 
construct explicitly. This suggests that lifting our first-order deformation construction
to all orders in ħ requires a homotopy-coherent generalization
of the concept of Drinfeld associators, involving suitable
hexagonators and pentagonators,  which we however currently 
do not know how to develop.


In Section <ref>, we apply our deformation construction 
from Section <ref> to explicit examples 
associated with L_∞-algebras. 
Following <cit.>, we describe
the formal classifying stack BL = [pt/L] of an L_∞-algebra L 
by its associated Chevalley-Eilenberg algebra ^∙(L).
For finite-dimensional L, this is a finitely generated semi-free CDGA,
hence our techniques from the previous sections apply to this case.
In particular, our main Theorem <ref>
shows that every 2-shifted Poisson structure
on ^∙(L) defines an explicit first-order braided monoidal deformation
of the homotopy 2-category 

    _^∙(L) ≃ 𝖧𝗈_2(𝐝𝐠𝐑𝐞𝐩(L))

of the symmetric monoidal dg-category of representations of the L_∞-algebra L
and their ∞-morphisms. In Subsection <ref>, we will specialize this result
to the case where L= is an ordinary
Lie algebra, which leads to a homotopical generalization of
the traditional constructions in <cit.> and <cit.>,
and in Subsection <ref> to the case where L=_κ is the string Lie 2-algebra
obtained from an ordinary Lie algebra  and a 3-cocycle κ:⋀^3→.


In the technical Appendix <ref>, we develop and present
a suitable higher-categorical framework in which our deformation
constructions in this paper are carried out. The main concept defined here
is the symmetric monoidal 3-category _R^[-1,0],ps from 
Proposition <ref> whose objects
are categories enriched over 2-term cochain complexes concentrated
in degrees -1 and 0, morphisms are enriched functors,
2-morphisms are enriched pseudo-natural transformations
and 3-morphisms are enriched modifications. Working with pseudo-natural transformations provides 
the required flexibility for the deformation constructions of this paper. Indeed, 
the homotopy 2-categories _A := 𝖧𝗈_2(_A^fgsf)
and their braided monoidal deformations from Section <ref>
are symmetric or, respectively, braided monoidal category objects in 
the symmetric monoidal 3-category _R^[-1,0],ps.








§ BASIC DERIVED ALGEBRAIC GEOMETRY




 §.§ Cochain complexes of modules


Throughout this work, we fix a field  of characteristic 0.
Let R be a commutative, associative and unital -algebra.
We denote the category of (not necessarily bounded) cochain complexes 
of R-modules by _R. Concretely, an object V=(V,_V)∈_R is a pair consisting of 
a family V={V^i}_i∈ of R-modules labeled by their degrees i∈
and a family _V = {_V^i : V^i → V^i+1}_i∈ of degree-increasing
R-linear maps (called the differential) which squares to zero, i.e. _V _V =0. A morphism f :V→ W in _R
is a family f={f^i : V^i→ W^i}_i∈ of degree-preserving R-linear maps
which commutes with the differentials, i.e. _W f = f _V.


The category _R is closed symmetric monoidal with respect to the following structures:
The tensor product V⊗_R W∈_R of two cochain complexes V,W∈_R is defined by
the relative tensor product of R-modules


(V⊗_R W)^i := ⊕_j∈ (V^j ⊗_R W^i-j)  ,

for all i∈, and the differential 

_V⊗_R W(v⊗_R w) :=  _V(v)⊗_R w + (-1)^|v| v⊗_R _W(w)  ,


for all homogeneous v∈ V and w∈ W, where | v|∈ denotes the degree of v.
The monoidal unit is R∈_R concentrated in degree 0 with trivial differential _R=0.
The symmetric braiding is defined by the Koszul sign rule

γ_V,W^ :  V⊗_R W ⟶ W⊗_R  V ,   v⊗_R w  ⟼ (-1)^|v| |w| w⊗_R v  ,

for all homogeneous v∈ V and w∈ W. The internal hom _R(V,W)∈_R between 
two cochain complexes V,W∈_R is defined by the internal hom of R-modules


_R(V,W)^i :=  ∏_j∈hom_R(V^j,W^j+i)  ,

for all i∈, and the differential 

__R(V,W)(f) := _W f - (-1)^|f | f _V  ,


for all homogeneous f∈_R(V,W).


In the special case where R= is the base field, we shall simply write
(_,⊗,,) for the closed symmetric monoidal structure, 
suppressing the subscript _ from our notations.


Given any integer n∈ and any cochain complex V∈_R, we define the n-shifted 
cochain complex V[n]∈_R by V[n]^i:= V^i+n, for all i∈, and _V[n]:= (-1)^n _V.
Using the identification V[n]≅ R[n]⊗_R V, we shall write elements of the n-shifted cochain
complex as s^-n v∈ V[n], where s^-n∈ R[n] denotes the degree -n element represented
by the unit of R.




 §.§ Semi-free CDGAs and dg-modules


A commutative differential graded algebra over the field  (in short, a CDGA) is a commutative
algebra object A = (A,_A,μ_A,η_A) in the closed symmetric monoidal category _.
More explicitly, a CDGA is a cochain complex A=(A,_A)∈_ 
endowed with _-morphisms μ_A : A⊗ A→ A (called multiplication)
and η_A : → A (called unit) which satisfy the usual associativity,
unitality and commutativity axioms internal to _.

A CDGA A= (A,_A,μ_A,η_A) is called semi-free
if its underlying -graded commutative algebra A^♯ = (A,μ_A,η_A) is free, 
i.e. 

A^♯ ≅ (V^♯)

is isomorphic to the symmetric algebra of a -graded vector space V^♯.
A semi-free CDGA is called finitely generated if the total dimension
(V^♯) := ∑_i∈(V^i)<∞ of the -graded 
vector space V^♯ is finite.


Associated with any CDGA A is its dg-category _A of (left) A-dg-modules.
More explicitly, an object M = (M,_M,ℓ_M) in this category is
a cochain complex M=(M,_M)∈_ endowed with a _-morphism 
ℓ_M : A⊗ M→ M (called left action) which satisfies the 
usual left module axioms internal to _. The cochain complex
of morphisms between two objects M,N∈_A is defined as the subcomplex


_A(M,N) ⊆  (M,N)

of the internal hom in _ containing the homogeneous elements
f∈(M,N) which commute with the left actions, i.e.
f(ℓ_M(a⊗m)) = (-1)^|f| |a| ℓ_N(a⊗f(m))  ,


for all a∈ A and m∈ M. The dg-category _A is closed symmetric monoidal
with respect to the relative tensor product

M⊗_A N := (
@C=8em
M⊗N       @<1ex>[l]^-𝕀_M⊗ℓ_N@<-1ex>[l]_-(ℓ_M⊗𝕀_N) (γ_M,A^⊗𝕀_N)M⊗A⊗N

) ∈ _A  ,

the monoidal unit A=(A,_A,μ_A)∈_A, the symmetric braiding

γ_M,N^ :  M⊗_A N  ⟶ N⊗_A M ,  m⊗_A n ⟼ (-1)^|m| |n| n⊗_A m  ,

for all homogeneous m∈ M and n∈ N, and the internal hom

_A(M,N) ∈ _A
given by the cochain complex (<ref>) and the left action 
A⊗_A(M,N)→_A(M,N) , a⊗ f ↦ a· f
defined by (a· f)(m) := ℓ_N(a⊗ f(m)), for all a∈ A,
f∈_A(M,N) and m∈ M.

An A-dg-module M = (M,_M,ℓ_M) is called semi-free
if its underlying A^♯-module M^♯ = (M,ℓ_M) is free, i.e. 

M^♯ ≅ A^♯⊗W^♯
for a -graded vector space W^♯. 
A semi-free A-dg-module is called finitely generated if the total dimension
(W^♯) := ∑_i∈(W^i)<∞ of the -graded 
vector space W^♯ is finite. We denote by

_A^fgsf ⊆  _A
the full dg-subcategory whose objects are all finitely generated 
semi-free A-dg-modules.



The closed symmetric monoidal structure on _A
restricts to a closed symmetric monoidal structure
on the full dg-subcategory _A^fgsf⊆_A.
Indeed, the monoidal unit A∈_A is finitely generated semi-free and,
given any two finitely generated semi-free A-dg-modules
M,N∈_A^fgsf with 
M^♯≅ A^♯⊗ W^♯ and N^♯≅ A^♯⊗ U^♯, 
we have isomorphisms


(M⊗_A N)^♯ ≅ 
A^♯⊗(W^♯⊗U^♯)

and

_A(M,N)^♯ ≅ A^♯⊗(U^♯⊗(W^♯)^∗)  ,


which imply that M⊗_A N∈_A^fgsf and
_A(M,N)∈_A^fgsf are finitely generated semi-free.
Here (W^♯)^∗ denotes the dual of the -graded vector space W^♯,
which exists since W^♯ is, by hypothesis, of finite total dimension.





 §.§ Derivations, Kähler differentials and shifted Poisson structures


We recall some basic aspects of the geometry of a finitely generated semi-free CDGA A, i.e.A^♯≅ (V^♯) for some -graded vector space V^♯ of finite total dimension.
The dg-module of derivations 𝖳_ A∈_A of A is defined as the subcomplex


𝖳_ A ⊆  (A,A)

of the internal hom in _ containing the homogeneous elements D∈(A,A) which satisfy the
Leibniz rule

D(a a^') =  D(a) a^'+ (-1)^|D|  |a| a D(a^')  ,


for all homogeneous a,a^'∈ A. The A-dg-module structure 
A⊗𝖳_ A→𝖳_ A , a⊗ D↦ a· D is given by
(a· D)(a^'):= a D(a^'), for all a,a^'∈ A and D∈𝖳_ A.
Since A is finitely generated semi-free, each derivation is completely 
specified by its action on the generators V^♯, which yields an isomorphism

(𝖳_ A)^♯ ≅ A^♯⊗(V^♯)^∗  ,

where (V^♯)^∗ denotes the dual of the -graded vector space V^♯.
Hence, 𝖳_ A∈_A^fgsf is a finitely generated semi-free A-dg-module.


The underlying cochain complex of 𝖳_ A can be endowed with the Lie algebra structure
given by the commutator

[ · , · ] : 𝖳_ A⊗𝖳_ A   ⟶  𝖳_ A
 ,  D⊗D^' ⟼ [D,D^'] := 
D D^'- (-1)^|D| |D^'| D^' D  ,

for all homogeneous D,D^'∈𝖳_ A.
The Lie algebra and A-dg-module structures on 𝖳_ A satisfy the compatibility condition

[D,a·D^'] =  D(a)·D^'+ (-1)^|D| |a| a·[D,D^']  ,

for all homogeneous D,D^'∈𝖳_ A and a∈ A.


As a consequence of our hypothesis that A is a finitely generated semi-free CDGA, 
the dg-module of derivations 𝖳_ A∈_A^fgsf is dualizable. We define
the dg-module of Kähler differentials by the internal hom


Ω_A := _A(𝖳_ A , A ) ∈  _A^fgsf  ,

which is finitely generated semi-free as a consequence of Remark <ref> and

(Ω_A)^♯ ≅ A^♯⊗(V^♯)^∗∗  ≅ A^♯⊗V^♯  .


We denote the duality pairings between Ω_A and 𝖳_ A by


 ·  ·  : Ω_A⊗_A 𝖳_ A ⟶ A ,  
ω⊗_A D ⟼ ωD := ω(D)

and 

 ·  ·  : 𝖳_ A ⊗_AΩ_A ⟶ A ,  
D⊗_A ω ⟼ Dω := (-1)^|D| |ω| ω(D)  ,


for all homogeneous ω∈Ω_A and D∈𝖳_ A. The de Rham differential
_ : A→Ω_A is defined by

D_(a) := D(a)  ,

for all a∈ A and D∈𝖳_ A. From this definition one directly
checks that _(a a^') = _(a)· a^' + a·_(a^'),
for all a,a^'∈ A, and that _Ω_A _ = _ _A.


A crucial preliminary concept for the definition of shifted Poisson structures
is that of shifted polyvectors.

Let n∈ be an integer. The CDGA of n-shifted polyvectors on
a finitely generated semi-free CDGA A is defined as the relative symmetric algebra

(A,n) :=  _A(𝖳_ A[-n-1]) =  ⊕_m≥0_A^m(𝖳_ A[-n-1])

of the (-n-1)-shift of the A-dg-module of derivations 𝖳_ A. The non-negative integer
m∈^≥ 0 in the direct sum decomposition in terms of symmetric powers is called
the weight of polyvectors.


The CDGA of n-shifted polyvectors (A,n) can be endowed with a canonical ℙ_n+2-algebra
structure (called Schouten-Nijenhuis bracket)

{ · , · } ∈ ((A,n)⊗(A,n),(A,n))^-n-1

which is defined on the generators by


{a,a^'}    := 0  ,

{s^n+1D,a}    := D(a)  ,

{s^n+1D,s^n+1D^'}    := (-1)^|D| (n+1) s^n+1[D,D^']  ,


for all homogeneous a,a^'∈ A and D,D^'∈𝖳_ A, where we recall that
D(a)∈ A denotes the evaluation of the derivation D on a∈ A 
and [D,D^']∈𝖳_ A denotes
the Lie bracket (<ref>) on 𝖳_ A. The bracket
{ · , · } is then extended to (A,n) by the defining
properties of a ℙ_n+2-algebra structure, i.e.


  (i) Antisymmetry: For all homogeneous P,Q∈(A,n),

{P,Q} = -(-1)^(n+1) (-1)^|P| |Q| {Q,P}  .



  (ii) Derivation property: For all homogeneous P,Q,R∈(A,n),

{P,Q R} = {P,Q} R + (-1)^(|P|-n-1)  |Q| Q {P,R}  .



  (iii) Jacobi identity: For all homogeneous P,Q,R∈(A,n),

{ P,{Q,R} } = (-1)^(|P|-n-1) (n+1) {{P,Q} ,R } + (-1)^(|P|-n-1) (|Q|-n-1) {Q,{P,R}}  .



Note that the Schouten-Nijenhuis bracket decreases the polyvector weight by 1, i.e. for P,Q∈(A,n)
of weight m_P and m_Q, the weight of {P,Q}∈(A,n) is m_P + m_Q -1.


The definition of n-shifted Poisson structures in 
<cit.> and <cit.> uses a completion
of the ℙ_n+2-algebra (A,n)
in order to avoid bounds on the weights of polyvectors.
The relevant definition is as follows.

The ℙ_n+2-algebra of completed n-shifted polyvectors
on a finitely generated semi-free CDGA A is defined by

(A,n) :=  ∏_m≥0 ^m_A(𝖳_ A[-n-1])  .

A completed n-shifted polyvector is thus a formal sum P = ∑_m≥ 0 P^(m)
of homogeneous weight components P^(m)∈^m_A(𝖳_ A[-n-1]).
The ℙ_n+2-algebra structure on (A,n) is given by


P Q    :=  ∑_m≥0(∑_k+l=m P^(k) Q^(l))  ,

{P,Q}    := ∑_m≥0(∑_k+l-1=m {P^(k),Q^(l)})  ,


which is well-defined because the weight is bounded from below and hence
the sums in the parentheses are finite.


We now define the key concept of an n-shifted Poisson structure following
<cit.>, see also <cit.>.

An n-shifted Poisson structure on a finitely generated semi-free CDGA A 
is a completed n-shifted polyvector

π = ∑_m≥2 π^(m) ∈ (A,n)^n+2

of degree n+2 and weight ≥ 2 which satisfies the Maurer-Cartan equation

_(π) + 12 {π,π} = 0  .



Decomposing the Maurer-Cartan equation into homogeneous weight components
one obtains a tower of conditions

_(π^(2))    = 0  ,

_(π^(3))+12 {π^(2),π^(2)}    = 0  ,

    ⋮

_(π^(m))+12 ∑_k+l-1=m{π^(k),π^(l)}    = 0  ,

for all m≥ 3. The first condition states that the bivector component
π^(2) is closed with respect to the differential _, i.e. it
defines an (n+2)-cocycle in ^2_A(𝖳_ A[-n-1]).
The second condition provides a homotopical relaxation of the usual condition
that a Poisson bivector has a trivial Schouten-Nijenhuis bracket with itself,
with the trivector π^(3) playing the role of a homotopy witnessing this condition. 
The higher weight components π^(m) 
for m≥ 4 describe a coherent tower of higher homotopies for this relaxation.








§ INFINITESIMAL 2-BRAIDINGS


The aim of this section is to study a class of first-order deformations,
into a braided monoidal category, of the symmetric monoidal category _A^fgsf
of finitely generated semi-free dg-modules over a finitely generated
semi-free CDGA A over the field . The precise context
in which our deformation constructions take place is given by the symmetric monoidal
3-category _R^[-1,0],ps from Proposition <ref>,
consisting of categories, functors, pseudo-natural transformations
and modifications that are suitably enriched over the symmetric monoidal category
^[-1,0]_R of 2-term cochain complexes concentrated in degrees -1 and 0. 
Our starting point is thus given by the homotopy 2-category

_A := 𝖧𝗈_2(_A^fgsf) ∈  ^[-1,0],ps_
from Appendix <ref>, which we endow with the symmetric monoidal
structure that is induced from the standard one on _A^fgsf given in
(<ref>) and (<ref>).


The effect of forming the homotopy 2-category
is that the cochain complex _A(M,N)∈_ of morphisms 
between any two dg-modules M,N∈_A^fgsf from (<ref>) 
gets replaced by its good truncation _A(M,N):=τ^[-1,0](_A(M,N))∈_^[-1,0]
to degrees -1 and 0. Concretely, this means that the homotopy 2-category 
_A = 𝖧𝗈_2(_A^fgsf)  describes only 
degree 0 morphisms between A-dg-modules which commute 
with the differentials and 1-homotopies (modulo 2-homotopies) between such morphisms,  
but it does not include information about higher homotopies. Our
motivation behind this truncation of the dg-category _A^fgsf 
is that the homotopy 2-category _A = 𝖧𝗈_2(_A^fgsf) 
is a 2-categorical object whose deformations can be described very explicitly 
in the context of the 3-category _R^[-1,0],ps from Proposition <ref>.
Most importantly, every tower of homotopy coherence data will automatically truncate beyond the level of 1-homotopies.
Our main result is that every 2-shifted Poisson structure on the CDGA A as in Definition <ref>
defines a very explicit first-order deformation of the braiding on _A.




 §.§ Setup and definitions


To set up a framework for studying first-order deformations
of the symmetric monoidal category object _A from (<ref>) in the symmetric 
monoidal 3-category _R^[-1,0],ps from Proposition <ref>,
we introduce a deformation parameter ħ which squares to zero ħ^2=0.
Performing a change of base along the 
commutative -algebra morphism →[ħ]/(ħ^2) defines
an object

_A^ħ :=  _A⊗[ħ]/(ħ^2) ∈  _[ħ]/(ħ^2)^[-1,0],ps  ,

which we endow with the symmetric monoidal structure that is induced from the one on _A.
More concretely, _A^ħ is the ^[-1,0]_[ħ]/(ħ^2)-enriched
category with the same objects as _A, i.e.  finitely generated
semi-free A-dg-modules, and whose morphisms 
are given by _A^ħ(M,N) = _A(M,N)⊗[ħ]/(ħ^2)∈^[-1,0]_[ħ]/(ħ^2),
for all M,N∈_A^ħ. Following the notation from Appendix <ref>,
we shall denote the symmetric braiding on _A^ħ by

    _A^ħ⊠_A^ħ      _A^ħ
       _A^ħ⊠_A^ħ[""name=0, anchor=center, inner sep=0, "⊗_A", from=1-1, to=1-3]
    	["τ"', from=1-1, to=2-2]
    	["⊗_A"', from=2-2, to=1-3]
    	["γ", shorten <=6pt, shorten >=3pt, Rightarrow, from=0, to=2-2]
      ,

where τ is an abbreviation for the component τ__A^ħ,_A^ħ
of the symmetric braiding in (<ref>) of the ambient 3-category 
_[ħ]/(ħ^2)^[-1,0],ps. 
When written in terms of the 3-categorical composition operations
from Appendix <ref>,
the symmetric braiding reads as γ: ⊗_A⇒⊗_A ∗τ.
Note that, by construction, this is a strict 
^[-1,0]_[ħ]/(ħ^2)-enriched natural transformation.


In the context of braided monoidal category
objects in _[ħ]/(ħ^2)^[-1,0],ps
from Definition <ref>, we would like to study deformations of
the braiding γ: ⊗_A⇒⊗_A ∗τ of _A^ħ
which keep all the other structures (i.e. the tensor product, monoidal unit, associator and unitors) 
fixed. The general ansatz for a first-order deformation of the 
braiding γ: ⊗_A⇒⊗_A ∗τ of _A^ħ
is obtained by considering a ^[-1,0]_[ħ]/(ħ^2)-enriched pseudo-natural transformation
of the form

γ^ħ :=  γ∘(+ ħ2  t) : 
⊗_A ⟹ ⊗_A ∗τ : _A^ħ⊠_A^ħ ⟶ _A^ħ
which satisfies the two hexagon identities



.9!@C=2em@R=2em   ⊗_A∗ (𝕀⊠⊗_A)@=>[r]^-γ^ħ∗   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_23∗τ_12@=>[dr]^-     α∗∗   
⊗_A∗ (⊗_A⊠𝕀) @=>[ru]^-α @=>[dr]_-∗ (γ^ħ⊠)              ⊗_A∗ (𝕀⊠⊗_A) ∗τ_23∗τ_12
   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_12@=>[r]_-α∗   ⊗_A∗ (𝕀⊠⊗_A) ∗τ_12@=>[ru]_-     ∗ (⊠γ^ħ)∗


and


.9!@C=2em@R=2em   ⊗_A∗ (⊗_A⊠𝕀)@=>[r]^-γ^ħ∗   ⊗_A∗ (𝕀⊠⊗_A)∗τ_12∗τ_23@=>[dr]^-     α^-1∗∗   
⊗_A∗ (𝕀⊠⊗_A) @=>[ru]^-α^-1 @=>[dr]_-∗ (⊠γ^ħ)              ⊗_A∗ (⊗_A⊠𝕀)∗τ_12∗τ_23   ,
   ⊗_A∗ (𝕀⊠⊗_A) ∗τ_23@=>[r]_-α^-1∗   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_23@=>[ru]_-     ∗ (γ^ħ⊠)∗



where α denotes the associator and we use 
the typical short-hand notations τ_12:= τ⊠𝕀 and 
τ_23:=𝕀⊠τ. Note that in the top horizontal
arrows of these diagrams we applied the 
properties (<ref>) and (<ref>)
of the symmetric braiding τ.
The composition of the  ^[-1,0]_[ħ]/(ħ^2)-enriched 
pseudo-natural transformations along the hexagons (<ref>) is given by the 
∘-composition from Appendix <ref> and these diagrams are required to
commute strictly, i.e. up to equality of enriched pseudo-natural transformations. 



Leveraging the fact that ħ^2=0, the hexagon identities are equivalent
to two linear relations for the deformation t, which can be regarded as 
the datum of a _^[-1,0]-enriched pseudo-natural
transformation (without change of base along →[ħ]/(ħ^2))
of the form t : ⊗_A ⇒⊗_A for the relative tensor
product on _A. 

The first hexagon identity (<ref>) for the deformed braiding (<ref>) 
is equivalent to the identity


    α^-1∘(t∗)∘α = ∗(t⊠) 
    
    + (∗(γ⊠)^-1)∘((α^-1∘(∗(⊠ t))∘α) ∗)∘(∗(γ⊠))

of _^[-1,0]-enriched 
pseudo-natural transformations 
⊗_A∗ (⊗_A⊠𝕀)⇒⊗_A∗ (⊗_A⊠𝕀).
The second hexagon identity (<ref>) for the deformed braiding (<ref>) 
is equivalent to the identity

    α∘(t∗)∘α^-1 = ∗(⊠ t) 
    
    + (∗(⊠γ)^-1)∘((α∘(∗(t⊠))∘α^-1) ∗)∘(∗(⊠γ))


of _^[-1,0]-enriched 
pseudo-natural transformations 
⊗_A∗ (𝕀⊠⊗_A)⇒⊗_A∗ (𝕀⊠⊗_A).


The proof follows directly by computing and simplifying the order ħ 
terms of the hexagon identities. The key observation is 
that the pseudo-functor coherences (<ref>)
which describe the interchange law between the ∗ and ∘ compositions
are trivial in all instances appearing in this calculation. Let us illustrate this
through an example: The order ħ term corresponding to the upper path of (<ref>)
involves the pseudo-natural transformation (γ∘ t)∗, which arises
from a particular composition of the pasting diagram

    _A𝐂⊠_A𝐂⊠_A𝐂      _A𝐂⊠_A𝐂      _A𝐂
             ["id⊠⊗_A", curve=height=-30pt, from=2-1, to=2-3]
    	["id⊠⊗_A"', curve=height=30pt, from=2-1, to=2-3]
    	[""name=0, anchor=center, inner sep=0, "id⊠⊗_A"description, from=2-1, to=2-3]
    	["⊗_A", curve=height=-30pt, from=2-3, to=2-5]
    	["⊗_A∗τ"', curve=height=30pt, from=2-3, to=2-5]
    	[""name=1, anchor=center, inner sep=0, "⊗_A"description, from=2-3, to=2-5]
    	["Id "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1-2, to=0]
    	["t "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1-4, to=1]
    	["Id "', shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=3-2]
    	["γ "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1, to=3-4]
        .

The pseudo-functor coherences (<ref>) for this pasting diagram are trivial
because of the identity t_(M,N⊗_A L),(M, N⊗_A L)(𝕀_(M,N⊗_A L)) =0,
for all (M,N,L)∈_A𝐂⊠_A𝐂⊠_A𝐂,
given by property (2) of the Definition <ref>
of _^[-1,0]-enriched pseudo-natural transformations.
It then follows that 

(γ∘t)∗ = (γ∘t)∗(∘)  = (γ∗)∘(t∗)   ,

which is the key step to identify the left-hand side of (<ref>). 
In addition to arguments of this form, one also has to use that 
the pseudo-functor coherences (<ref>) are trivial
whenever ζ^' is strictly natural, i.e. ζ^'_d,d^'=0 for all d,d^'∈.


This result justifies the following 

An infinitesimal 2-braiding on the symmetric monoidal category object
_A=𝖧𝗈_2(_A^fgsf) in the symmetric monoidal 3-category
^[-1,0],ps_ from Proposition <ref>
is a _^[-1,0]-enriched pseudo-natural transformation 
t: ⊗_A⇒⊗_A : _A⊠_A→_A which satisfies 
the two infinitesimal hexagon identities in (<ref>).



As a consequence of Proposition <ref>, the datum of an infinitesimal
2-braiding t is equivalent to a first-order deformation
γ^ħ = γ∘ ( +ħ2 t) : ⊗_A⇒⊗_A∗τ of the symmetric braiding
on _A to a braiding on _A^ħ = _A⊗[ħ]/(ħ^2).
Note that the deformed braiding is not necessarily symmetric because

(γ^ħ∗)∘γ^ħ = + ħ2 ( t + (γ∗)∘(t∗)∘γ) 
 :  ⊗_A ⟹ ⊗_A

is in general different from .



Using the explicit composition formulas from Appendix <ref>, we can 
express the infinitesimal hexagon identities (<ref>)
in terms of the components of the _^[-1,0]-enriched pseudo-natural transformations involved.
(Recall also Definition <ref>.) In this calculation,
various simplifications arise from the fact that all transformations but t
are strictly natural, i.e. their homotopy data given by the double-indexed components
(<ref>) is trivial. To ease notations, we suppress 
the components of the associator α in what follows. One then finds that the first
infinitesimal hexagon identity (<ref>) is equivalent to


t_M,N⊗_A L =  t_M,N⊗_A 𝕀_L +  
(γ_N,M⊗_A𝕀_L) (𝕀_N⊗_A t_M,L) (γ_M,N⊗_A𝕀_L)
  ,

for all M,N,L∈_A, and

    t_(M,N⊗_A L),(M^',N^'⊗_A L^')(h⊗(k⊗_A l)) = 
    t_(M,N),(M^',N^')(h⊗k)⊗_A l 
    
    +(γ_N^',M^'⊗_A𝕀_L^') (k ⊗_A t_(M,L),(M^',L^')(h⊗ l))
     (γ_M,N⊗_A𝕀_L)
      ,


for all M,N,L,M^',N^',L^'∈_A, h∈_A(M,M^')^0, k∈_A(N,N^')^0
and l∈_A(L,L^')^0.
The second infinitesimal hexagon identity (<ref>) is equivalent to


t_M⊗_A N,L =  𝕀_M⊗_A t_N,L + 
(𝕀_M⊗_Aγ_L,N) (t_M,L⊗_A𝕀_N)
 (𝕀_M⊗_Aγ_N,L)
  ,

for all M,N,L∈_A, and

    t_(M⊗_A N,L),(M^'⊗_A N^',L^')((h⊗_A k)⊗l) = 
    h⊗_A t_(N,L),(N^',L^')(k⊗l) 
    
    +(𝕀_M^'⊗_Aγ_L^',N^') (t_(M,L),(M^',L^')(h⊗l)⊗_A k)
     (𝕀_M⊗_Aγ_N,L)
      ,


for all M,N,L,M^',N^',L^'∈_A, h∈_A(M,M^')^0, k∈_A(N,N^')^0
and l∈_A(L,L^')^0.


Our examples of infinitesimal 2-braidings in Subsection <ref>
will satisfy a useful additional property that simplifies their analysis.

An infinitesimal 2-braiding t: ⊗_A⇒⊗_A is 
called γ-equivariant if it satisfies the identity

γ∘t = (t∗)∘γ
of _^[-1,0]-enriched pseudo-natural transformations ⊗_A⇒⊗_A∗τ.



It is important to emphasize that γ-equivariance does not imply
that the associated deformed braiding γ^ħ is symmetric. Indeed,
recalling the calculation in Remark <ref>, we find that

(γ^ħ∗)∘γ^ħ = + ħ  t  ,

for every γ-equivariant infinitesimal 2-braiding t.



Let t: ⊗_A⇒⊗_A be any _^[-1,0]-enriched pseudo-natural transformation
which satisfies the γ-equivariance property (<ref>).
Then the two infinitesimal hexagon identities (<ref>) for t are equivalent.


This can be verified directly by using the equivalent component-wise
description of the infinitesimal hexagon identities from Remark <ref>
and the fact that the γ-equivariance property reads in components as


γ_M,N t_M,N  =  t_N,M γ_M,N  ,

for all M,N∈_A, and

γ_M^',N^' t_(M,N),(M^',N^')(h⊗k)  = 
t_(N,M),(N^',M^')(k⊗h) γ_M,N  ,


for all
M,N,M^',N^'∈_A, h∈_A(M,M^')^0 and k∈_A(N,N^')^0.






 §.§ Construction from 2-shifted Poisson structures


Suppose that π = ∑_m≥ 2π^(m)∈(A,2)^4
is a 2-shifted Poisson structure (recall Definition <ref>)
on the finitely generated semi-free CDGA A. 
In this subsection, we will
present an explicit construction which assigns to
this datum a γ-equivariant infinitesimal 2-braiding t:⊗_A⇒⊗_A
in the sense of Definitions <ref>
and <ref>
on the homotopy 2-category _A =𝖧𝗈_2(_A^fgsf)∈_^[-1,0],ps of the dg-category of finitely generated semi-free A-dg-modules.
We will illustrate later in Section <ref>
that our construction is a non-trivial homotopical generalization
of the standard construction of an infinitesimal braiding on the representation category
𝐑𝐞𝐩() of a Lie algebra  from an invariant symmetric tensor in (^2)^, see
e.g. <cit.> and <cit.>.


Our infinitesimal 2-braiding t is built as a composite of 
more primitive _^[-1,0]-enriched pseudo-natural transformations
which we are going to define now. Let us denote by

Ω_A[1]⊗_A( · )  :  _A ⟶ _A
the _^[-1,0]-enriched functor of taking the relative tensor product 
with the 1-shift Ω_A[1]∈_A of the dg-module of Kähler differentials
from (<ref>). We define a 
_^[-1,0]-enriched pseudo-natural transformation


ξ :  𝕀 ⟹ Ω_A[1]⊗_A( · ) :  _A ⟶ _A
from the _^[-1,0]-enriched  identity functor 𝕀 : _A→_A
by specifying its components as in Definition <ref>:
For each object M∈_A, the component 

(ξ_M : M→Ω_A[1]⊗_A M) ∈  _A(M,Ω_A[1]⊗_A M)^0

is the A-dg-module map 
which is defined on the generators ⊗ w∈ M^♯≅ A^♯⊗ W^♯ 
of M by

ξ_M(⊗w) :=  s^-1 (_⊗𝕀)(_M(⊗w))  ,

where we recall our notation from Subsection <ref>
that s^-1∈[1] denotes the degree -1 element represented by the unit of .
For each pair of objects M,M^'∈_A and each A-dg-module map 
(h: M→ M^')∈_A(M,M^')^0,
the component 

(ξ_M,M^'(h) : M→Ω_A[1]⊗_A M^') ∈  
_A(M,Ω_A[1]⊗_A M^')^-1

is the homotopy which is defined on the generators
⊗ w∈ M^♯≅ A^♯⊗ W^♯ 
of M by

ξ_M,M^'(h)(⊗w) := - s^-1 (_⊗𝕀)(h(⊗w))  .



The components in (<ref>) define a _^[-1,0]-enriched 
pseudo-natural transformation ξ : 𝕀⇒Ω_A[1]⊗_A( · ).


Before we verify the three properties (1), (2) and (3) from Definition <ref>,
let us first comment on the well-definedness of the components of ξ.
Concerning (i) from Definition <ref>, 
we have to show that __A(ξ_M) := _Ω_A[1]⊗_A M ξ_M - ξ_M _M 
= 0, i.e. that ξ_M commutes with the differentials, which can be verified by picking a basis
{w_i∈ W^♯} and evaluating on the associated generators ⊗ w_i∈ M. 
This allows us to write _M(⊗ w_i) =: M_i^ j⊗ w_j
and hence ξ_M(⊗ w_i) = s^-1 _(M_i^ j)⊗ w_j, 
with M_i^ j∈ A and summations over repeated indices understood. We then compute

__A(ξ_M)(⊗w_i)    = 
_Ω_A[1]⊗_A M(ξ_M(⊗w_i)) - 
ξ_M(_M(⊗w_i) )

    =  -s^-1_Ω_A⊗_A M(_(M_i^ j)⊗w_j)
- M_i^ j s^-1 _(M_j^ k)⊗w_k

     = -s^-1 (_(_A(M_i^ j)) ⊗w_j
+ (-1)^|M_i^ j|  _( M_i^ j M_j^ k)⊗w_k )

    = 0  ,

where the last identity follows from expressing the square-zero condition _M_M(⊗ w_i)=0
in our index notation. Concerning (ii) from Definition <ref>, 
we note that ξ_M,M^'(h) as defined 
in (<ref>) is clearly -linear in the argument h.


Property (2) from Definition <ref> follows immediately
from _=0. To verify the first property in (1),
we use also the index notations h(⊗ w_i) =:h_i^ j⊗ w^'_j
and _M^'(⊗ w^'_i)=:M_i^' j⊗ w^'_j,
where {w^'_i∈ W^'♯} is a choice of basis for the generators
of M^', and compute

__A(ξ_M,M^'(h))(⊗w_i)    = 
_Ω_A[1]⊗_A M^'(ξ_M,M^'(h)(⊗w_i))
+ ξ_M,M^'(h)(_M(⊗w_i))

     = s^-1 (_Ω_A⊗_A M^'(_(h_i^ j)⊗w^'_j)
- M_i^ j _(h_j^ k)⊗w^'_k)

     = s^-1 (_(_A(h_i^ j))⊗w^'_j
+ (-1)^|h_i^ j| _(h_i^ j) M_j^' k⊗w^'_k

           - M_i^ j _(h_j^ k)⊗w^'_k)

    = s^-1(- (-1)^|h_i^ j| h_i^ j _(M_j^' k)⊗w^'_k
+ _(M_i^ j) h_j^ k⊗w^'_k)

    =  (𝕀_Ω_A[1] ⊗_A h)(ξ_M(⊗w_i)) - 
ξ_M^'(h(⊗w_i))  ,

where the fourth step follows from expressing
the cocycle condition __A(h)(⊗ w_i)=0 in our index notation.
To verify the second property in (1), one performs a similar calculation for 
(k: M→ M^')∈_A(M,M^')^-1 of degree -1 and finds that 

   (𝕀_Ω_A[1] ⊗_A k)(ξ_M(⊗w_i)) - 
ξ_M^'(k(⊗w_i)) -ξ_M,M^'(__A(k))(⊗w_i)

      =  s^-1(_(_A(k_i^ j))⊗w_j^'+ 
(-1)^|k_i^ j|  _(k_i^ j)  M_j^' k⊗w_k^'+ 
(-1)^|M_i^ j|  M_i^ j _(k_j^ k)⊗w^'_k)  .

The right-hand side can be identified as the differential __A(q)(⊗ w_i) of the 
2-homotopy q∈_A(M,Ω_A[1]⊗_A M^')^-2 which is defined on generators by
q(⊗ w) = -s^-1 (_⊗𝕀)(k(⊗ w)),
hence the right-hand side is zero in our truncated cochain complex _A(M,Ω_A[1]⊗_A M^').


It remains to verify property (3) from Definition <ref>.
Given (h: M→ M^')∈_A(M,M^')^0 and (h^' : M^'→ M^'')
∈_A(M^',M^'')^0, we introduce one more index notation
h^'(⊗ w_i^')=: h_i^' j⊗ w_j^'', where 
{w^''_i∈ W^''♯} is a choice of basis for the generators
of M^'', and compute

ξ_M,M^''(h^' h)(⊗w_i)    = 
-s^-1 _(h_i^ j h_j^' k)⊗w_k^''

    = -s^-1(_(h_i^ j) h_j^' k⊗w_k^'' +
h_i^ j _(h_j^' k)⊗w_k^'')

    = ξ_M^',M^''(h^')(h(⊗w_i)) + 
(𝕀_Ω_A[1] ⊗_A h^')(ξ_M,M^'(h)(⊗w_i))  .

This concludes the proof.


In addition to the _^[-1,0]-enriched pseudo-natural transformation
ξ from (<ref>), we require two more strict
_^[-1,0]-enriched natural transformations to build our infinitesimal 2-braiding.
First, using the original symmetric braiding γ, we obtain a family of isomorphisms


𝕀⊗_Aγ_M,Ω_A[1]⊗_A𝕀 : Ω_A[1]⊗_A M ⊗_A Ω_A[1]⊗_A N
 ⟶  Ω_A[1]⊗_A Ω_A[1]⊗_A M ⊗_A N

in _A, for all M,N∈_A, which 
defines a strict _^[-1,0]-enriched natural isomorphism

⊗_A ∗((Ω_A[1]⊗_A( · ))⊠(Ω_A[1]⊗_A( · )) )
 ≅⟹ (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A  .


Second, consider the extension 
⟨ · , · ⟩ :_A^2(𝖳_ A[-1])⊗_A Ω_A[1]⊗_AΩ_A[1]
→ A of the duality pairing (<ref>) defined by

    ⟨ sD sD^', s^-1ω⊗_A s^-1ω^'⟩ := 
    (-1)^|ω| + | D| ⟨ D, ⟨ D^',ω⟩·ω^'⟩
    
    + (-1)^(| D| +1) (| D^'| +1) (-1)^|ω| + | D^'| ⟨ D^', ⟨ D,ω⟩·ω^'⟩  ,
 
for all homogeneous D, D^'∈𝖳_ A 
and ω,ω^'∈Ω_A.
Considering the weight 2 component 
π^(2)∈_A^2(𝖳_ A[-3])^4≅_A^2(𝖳_ A[-1])^0
of the 2-shifted Poisson structure (recall that this is a cocycle, see Remark <ref>),
we obtain a family of morphisms


⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀 :  Ω_A[1]⊗_AΩ_A[1]⊗_A M⊗_A N ⟶ A⊗_A M⊗_A N  ≅ 
M⊗_A N

in _A, for all M,N∈_A, which defines a 
strict _^[-1,0]-enriched natural transformation

ι_π :  (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A ⟹ 
⊗_A  .


Using the ∘-composition of 
_^[-1,0]-enriched pseudo-natural transformations from Appendix <ref>,
we define the _^[-1,0]-enriched pseudo-natural transformation 


.9!@C=0.1em@=[d]⊗_A@==>[rr]^-t          ⊗_A
⊗_A∗(𝕀⊠𝕀) @=>[dr]_-∗ (ξ⊠ξ)              (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A@=>[u]_-ι_π
     ⊗_A ∗((Ω_A[1]⊗_A( · ))⊠(Ω_A[1]⊗_A( · ))) @=>[ru]_-  ≅


as our candidate for an infinitesimal 2-braiding on _A.
Note that this candidate depends only on the bivector component
π^(2) of the 2-shifted Poisson structure π = ∑_m≥ 2π^(m)∈(A,2)^4.

Given any 2-shifted Poisson structure π∈(A,2)^4 on a finitely generated
semi-free CDGA A over the field , the _^[-1,0]-enriched pseudo-natural transformation t 
from (<ref>) defines a γ-equivariant
infinitesimal 2-braiding in the sense of Definitions <ref>
and <ref>
on the homotopy 2-category _A =𝖧𝗈_2(_A^fgsf)∈_^[-1,0],ps of the dg-category of finitely generated semi-free A-dg-modules.


The most straightforward proof of this result is to compute
the components of the _^[-1,0]-enriched pseudo-natural transformation t
from (<ref>), and then to verify directly that 
these components satisfy the γ-equivariance property from (<ref>)
and the infinitesimal hexagons from Remark <ref>.
Using the composition operations from Appendix <ref>, one finds
by a straightforward computation that



@C=6.5em@R=3em
M⊗_A N [d]_-ξ_M⊗_A ξ_N [r]^-t_M,N     M⊗_A N 

Ω_A[1]⊗_A M⊗_A Ω_A[1]⊗_A N  [r]_-𝕀⊗_Aγ_M,Ω_A[1]⊗_A 𝕀     
Ω_A[1]⊗_A Ω_A[1]⊗_A M⊗_A N [u]_-⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀

    ,

for all M,N∈_A, and

.9!@C=6.5em@R=4em
M⊗_A N [d]_-ξ_M,M^'(h)⊗_A (𝕀⊗_A k)ξ_N
+ ξ_M^'h⊗_A ξ_N,N^'(k)[r]^-t_(M,N),(M^',N^')(h⊗k)     M^'⊗_A N^'
Ω_A[1]⊗_A M^'⊗_A Ω_A[1]⊗_A N^'[r]_-𝕀⊗_Aγ_M^',Ω_A[1]⊗_A 𝕀     Ω_A[1]⊗_A Ω_A[1]⊗_A M^'⊗_A N^'[u]_-⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀    ,


for all M,N,M^',N^'∈_A, h∈_A(M,M^')^0 and k∈_A(N,N^')^0.


The γ-equivariance property (<ref>)
for the degree 0 components t_M,N follows immediately by using 
symmetry of the extended pairing (<ref>), i.e.⟨π^(2), · ⟩ γ  = ⟨π^(2), · ⟩,
and the hexagon identities for the original symmetric braiding γ.
To prove the γ-equivariance property (<ref>)
for the degree -1 components t_(M,N),(M^',N^')(h⊗k),
one has to use additionally that

   γ_Ω_A[1]⊗_A M^', Ω_A[1]⊗_A N^' (ξ_M,M^'(h)⊗_A (𝕀⊗_A k)ξ_N+ ξ_M^'h⊗_A ξ_N,N^'(k)) γ_N,M

         =  (𝕀⊗_A k)ξ_N ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A ξ_M^'h

         = ξ_N^' k ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A (𝕀⊗_A h) ξ_M + __A(ξ_N,N^'(k)⊗_Aξ_M,M^'(h))

         = ξ_N^' k ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A (𝕀⊗_A h) ξ_M   ,

where in the second step we used property (1) of the Definition <ref>
of _^[-1,0]-enriched pseudo-natural transformations and in the last step
we used that the morphism complexes of _A are by definition truncated to degrees {-1,0},
hence the 2-homotopy in the last term gets identified with 0.


It remains to verify the first infinitesimal hexagon identity (<ref>),
since the second one (<ref>) then 
follow from Proposition <ref> and γ-equivariance.
For this we recall the definition of the components of ξ from (<ref>)
and observe that


ξ_N⊗_A L  =  ξ_N⊗_A𝕀+ (γ_N,Ω_A[1]⊗_A𝕀) (𝕀⊗_Aξ_L)  ,

for all N,L∈_A, and

ξ_N⊗_A L,N^'⊗_A L^'(k⊗_A l) =  
 ξ_N,N^'(k)⊗_A l + (γ_N^',Ω_A[1]⊗_A𝕀) (k⊗_Aξ_L,L^'(l))  ,


for all N,L,N^',L^'∈_A, k∈_A(N,N^')^0 and l∈_A(L,L^')^0.
Using also the hexagon identities for the original symmetric braiding γ,
one verifies the infinitesimal hexagon identity (<ref>).



For later use in our discussion of examples in 
Section <ref>, let us expand the components
t_M,N and t_(M,N),(M^',N^')(h⊗k)
of the infinitesimal 2-braiding (<ref>)
in terms of a choice of bases for the finitely generated
semi-free A-dg-modules M,N,M^',N^'∈_A.
Using similar notations as in the proof of Lemma <ref>,
we denote by {w_i∈ W^♯} a basis for M^♯≅ A^♯⊗ W^♯, 
by {u_q∈ U^♯} a basis for N^♯≅ A^♯⊗ U^♯,
by {w_i^'∈ W^'♯} a basis for M^'♯≅ A^♯⊗ W^'♯, 
and by {u_q^'∈ U^'♯} a basis for N^'♯≅ A^♯⊗ U^'♯.
Making use of these bases, we write


    _M(⊗ w_i)     = M_i^ j⊗ w_j  ,    _N(⊗ u_q)     = N_q^ r⊗ u_r  ,
    _M^'(⊗ w_i^') 
        = M_i^' j⊗ w_j^'  ,    _N^'(⊗ u_q^')
        = N_q^' r⊗ u_r^'  ,

for the differentials, 
and for the A-linear maps h∈_A(M,M^')^0 and 
k∈_A(N,N^')^0
we write 

h(⊗w_i)  =  h_i^ j⊗w_j^'  ,          k(⊗u_q)   =  k_q^ r⊗u_r^'  ,


where again all summations over repeated indices
are suppressed. Evaluating the diagram (<ref>), we then find the explicit basis
expression


   t_M,N(⊗w_i⊗u_q) =  (t_M,N)_iq^  jr⊗w_j⊗u_r  

        = (-1)^|w_j| (|N_q^ r|-1) 
⟨π^(2), s^-1_(M_i^ j)⊗_A s^-1_(N_q^ r)⟩⊗w_j⊗u_r  .

Furthermore, evaluating the diagram (<ref>), we find

   t_(M,N),(M^',N^')(h⊗k)(⊗w_i⊗u_q) = 
(t_(M,N),(M^',N^')(h⊗k))_iq^  jr⊗w_j^'⊗u_r^'

        = -(-1)^|w_j^'| (|N_q^ s|+ |k_s^ r|-1) 
⟨π^(2), s^-1_(h_i^ j) ⊗_A s^-1_(N_q^ s) k_s^ r⟩⊗w_j^'⊗u_r^'

            -(-1)^|h_i^ k|+ |w_j^'| (|k_q^ r|-1)
 ⟨π^(2), s^-1h_i^ k _(M_k^' j)⊗_A s^-1_(k_q^ r)⟩⊗w_j^'⊗u_r^'  .


Given a concrete example for the bivector
π^(2)∈_A^2(𝖳_ A[-3])^4≅_A^2(𝖳_ A[-1])^0,
one can evaluate these expressions further by using the definition (<ref>)
of the duality pairing.




 §.§ Towards higher-order deformations


A powerful result in the 1-categorical theory of infinitesimal braidings
is that, upon the choice of a Drinfeld associator <cit.>, every infinitesimal
braiding lifts to a formal deformation (i.e. to all orders in ħ)
of the given symmetric monoidal category into a braided monoidal one.
See e.g. <cit.> and <cit.> for the details.
The aim of this subsection is to explore which aspects of this construction admit 
a generalization to our concept of infinitesimal 2-braidings on _A.


To study formal deformations to all orders in ħ, we introduce
the commutative -algebra [[ħ]] of formal power series and perform
a base change 

_A[[ħ]] := _A⊗[[ħ]] ∈ _[[ħ]]^[-1,0],ps

along the commutative -algebra morphism →[[ħ]].
The goal is then to deform the symmetric monoidal structure on _A[[ħ]]
into a braided monoidal structure. In order to simplify notations,
we suppress in what follows the original associator α and simply write

⊗_A^2  :=  ⊗_A∗(⊗_A⊠𝕀)  ≅  ⊗_A∗(𝕀⊠⊗_A)  :  
_A[[ħ]]⊠_A[[ħ]]⊠_A[[ħ]] ⟶  _A[[ħ]]

for the 2-times iterated tensor product, and similarly
⊗_A^n : _A[[ħ]]^⊠(n+1)→_A[[ħ]] for the n-times iterated tensor product.
Given any γ-equivariant infinitesimal 2-braiding t : ⊗_A⇒⊗_A,
we denote by


t_ij : ⊗_A^n ⟹ ⊗_A^n :   _A[[ħ]]^⊠(n+1) ⟶  _A[[ħ]]

the _[[ħ]]^[-1,0]-enriched pseudo-natural transformation which evaluates
t on the i-th and j-th tensor factor, for all i,j∈{1,…,n+1} with i≠ j.
Explicitly, the components of t_ij read as


@C=13em
[d]_-≅⊗_k=1^n+1 M_k [r]^-(t_ij)_M_1,…,M_n+1      ⊗_k=1^n+1 M_k

M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[r]_-t_M_i,M_j⊗_A⊗_k𝕀_M_k     
M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[u]_-≅

  ,

for all M_1,…,M_n+1∈_A[[ħ]], and


@C=13em
[d]_-≅⊗_k=1^n+1 M_k [r]^-(t_ij)_(M_1,…,M_n+1),(M_1^',…,M_n+1^')(h_1⊗⋯⊗h_n+1)      ⊗_k=1^n+1 M^'_k

M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[r]_-t_(M_i,M_j),(M_i^',M_j^')(h_i⊗h_j)⊗_A⊗_k h_k     
M^'_i⊗_A M^'_j⊗_A ⊗_k=1
k≠i,j^n+1 M^'_k[u]_-≅

  ,


for all M_1,…,M_n+1,M^'_1,…,M^'_n+1∈_A[[ħ]]
and h_k∈_A[[ħ]](M_k,M_k^')^0, for k=1,…,n+1,
where the unlabeled vertical isomorphisms are given by the original symmetric braiding γ.
Let us note that there is an obvious generalization
of the above construction which assigns to every pair 
I=(i_1⋯ i_I) and J=(j_1⋯ j_J) of tuples of pairwise distinct indices in {1,…,n+1}
the _[[ħ]]^[-1,0]-enriched pseudo-natural transformation 
t_IJ : ⊗_A^n ⟹ ⊗_A^n which evaluates t on the 
tensor product of the I-factors and the tensor product of the J-factors.
In this compact notation, the infinitesimal hexagons from (<ref>) 
and the γ-equivariance property from Definition <ref>
read as

t_i(jk) = t_ij + t_ik  ,    t_(ij)k =  t_ik + t_jk  ,    t_ij  = t_ji   ,

for all pairwise distinct i,j,k∈{1,…,n+1}.
Furthermore, one verifies that 

[t_ij,t_kl] := t_ij∘t_kl - t_kl∘t_ij = 0  ,

for all pairwise distinct i,j,k,l∈{1,…,n+1}, by explicitly computing the components.


Following the same strategy as in <cit.> and <cit.>, 
we consider a candidate for a deformed associator and braiding
on _A[[ħ]] which is given by the ansatz


α^ħ :=  Φ(t_12,t_23) : ⊗_A^2 ⟹ ⊗_A^2 :  
_A[[ħ]]⊠_A[[ħ]]⊠_A[[ħ]] ⟶ _A[[ħ]]

and

γ^ħ :=γ∘e^ħ/2 t :  
⊗_A ⟹ ⊗_A∗τ :  
_A[[ħ]]⊠_A[[ħ]] ⟶ _A[[ħ]]  ,


where Φ(𝔞,𝔟) is some formal power series in 
two non-commutative variables 𝔞 and 𝔟 satisfying Φ(0,0)=1.
In these expressions, both Φ and the exponential function e, which we describe by its formal Taylor series,
are evaluated by using the ∘-composition of pseudo-natural transformations from 
Appendix <ref>, hence α^ħ and γ^ħ are by construction
_[[ħ]]^[-1,0]-enriched pseudo-natural transformations.

The tuple (_A[[ħ]],⊗_A,A,α^ħ,λ,ρ,γ^ħ),
with deformed associator α^ħ and braiding γ^ħ 
from (<ref>), defines a semi-strict braided monoidal category 
object in _[[ħ]]^[-1,0],ps  in the sense of Definition <ref>
if and only if all of the following hold true:


  (1) The identity

Φ(t_12,t_23+ t_24)∘Φ(t_13+t_23,t_34) = 
Φ(t_23,t_34)∘Φ(t_12+t_13,t_24+t_34)∘Φ(t_12,t_23)

holds true in the algebra (PsNat(⊗_A^3,⊗_A^3),∘,_⊗_A^3)
of _[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.


  (2) The identities


Φ(t_23,t_13)∘e^ħ/2(t_12+t_13)∘Φ(t_12,t_23)
 =  e^ħ/2t_13∘Φ(t_12,t_13)∘e^ħ/2t_12

and

Φ(t_13,t_12)^-1 ∘e^ħ/2(t_13+t_23)∘Φ(t_12,t_23)^-1
 =  e^ħ/2t_13∘Φ(t_13,t_23)^-1∘e^ħ/2t_23


hold true in the algebra (PsNat(⊗_A^2,⊗_A^2),∘,_⊗_A^2)
of _[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.

Assuming additionally that Φ satisfies the inversion property 
Φ(𝔞,𝔟)^-1=Φ(𝔟,𝔞), then the two identities
in (<ref>) are equivalent.


The triangle identity holds true automatically as a consequence of Φ(0,0)=1
and the fact that all components of t involving the monoidal unit A∈_A[[ħ]] at least once 
are necessarily trivial. (The latter is a consequence of (<ref>).) Using (<ref>)
and arguments for strictness of the interchange law for ∘ and ∗
which are similar to the ones in the proof of Proposition <ref>, one shows that (1) is equivalent 
to the pentagon identity for α^ħ. With the same kind of computation,
one shows that (2) is equivalent to the two hexagon identities for γ^ħ and α^ħ.
Finally, under the additional assumption Φ(𝔞,𝔟)^-1=Φ(𝔟,𝔞),
one easily shows that (<ref>) and (<ref>) are identified
by permuting the indices according to 123↦ 321 and using once again (<ref>).



From a superficial perspective, the identities (1) and (2) in Proposition <ref>
take exactly the same form as the defining properties of a Drinfeld associator Φ,
see e.g. <cit.> or <cit.>.
However, there is the following caveat: The  analogous identities for a Drinfeld associator
are formulated in the universal enveloping algebra of the appropriate Drinfeld-Kohno Lie algebra
where the symbols t_ij satisfy the infinitesimal braid relations
t_ij = t_ji, [t_ij,t_kl]=0 and [t_ij,t_ik+t_jk]=0, for all pairwise distinct indices i,j,k,l.
In contrast to this, our identities
(1) and (2) must hold true in the appropriate algebra of 
_[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.
From (<ref>) and (<ref>),
we know that our t_ij's satisfy the first two types of infinitesimal braid relations,
i.e. t_ij = t_ji and [t_ij,t_kl]=0 for all pairwise distinct indices i,j,k,l,
but as we shall illustrate below they do not necessarily satisfy the third
type of relation [t_ij,t_ik+t_jk]=0 strictly. 


So the conclusion at this moment is as follows:
Assuming that our γ-equivariant infinitesimal 2-braiding t satisfies 
the remaining infinitesimal braid relations [t_ij,t_ik+t_jk]=0 strictly, then the choice of 
a Drinfeld associator Φ (in the traditional sense) provides
via Proposition <ref> a semi-strict
braided monoidal deformation (to all orders in ħ) 
of the symmetric monoidal category object (_A,⊗_A,A,α,λ,ρ,γ).


In light of the previous remark, we make the following observation.

The pseudo-naturality structure of the γ-equivariant infinitesimal 2-braiding
t:⊗_A⇒⊗_A induces _[[ħ]]^[-1,0]-enriched modifications


Γ_ijk : [t_ij,t_ik+t_jk]      0 :  ⊗_A^n ⟹ ⊗_A^n : 
_A[[ħ]]^⊠(n+1) ⟶ _A[[ħ]]  ,

for all pairwise distinct i,j,k∈{1,…,n+1}, which enforce weakly the remaining infinitesimal braid relations.
The components (recall Definition <ref>) of these modifications are given explicitly by

.9!@C=17em[d]_-≅⊗_l=1^n+1 M_l [r]^-(Γ_ijk)_M_1,…,M_n+1     ⊗_l=1^n+1 M_l

M_i⊗_A M_j⊗_A M_k⊗_A ⊗_l=1
l≠ i,j,k^n+1 M_l
[r]_-t_(M_i⊗_A M_j,M_k),(M_i⊗_A M_j,M_k) (t_M_i,M_j⊗𝕀)⊗_A𝕀     
M_i⊗_A M_j⊗_A M_k⊗_A ⊗_l=1
l≠ i,j,k^n+1 M_l[u]_-≅  ,


for all M_1,…,M_n+1∈_A[[ħ]]. 


Recalling (<ref>), one observes that 

[t_ij,t_ik+t_jk]  =  [t_ij,t_(ij)k]  =  t_ij∘t_(ij)k -t_(ij)k∘t_ij  .

Using the composition formulas from Appendix <ref>, the component expressions
(<ref>) and pseudo-naturality of t_(ij)k in the left entry, one computes 
the components of this _[[ħ]]^[-1,0]-enriched pseudo-natural 
transformation and finds that 

(t_ij∘t_(ij)k -t_(ij)k∘t_ij)_M_1,…,M_n+1  = __A((Γ_ijk)_M_1,…,M_n+1)

with Γ_ijk defined in (<ref>). By a routine calculation
one then shows that Γ_ijk satisfies the properties of a 
_[[ħ]]^[-1,0]-enriched modification from Definition <ref>.



The reader might have noticed that there are potential ambiguities
in defining the _[[ħ]]^[-1,0]-enriched modifications Γ_ijk.
These arise from the multiple equivalent ways to write the expression [t_ij,t_ik+t_jk]
by using the identities in (<ref>). 
Such ambiguities do not seem to be present for our γ-equivariant infinitesimal 2-braidings.
Indeed, one observes that

Γ_ijk  = Γ_jik

is symmetric in the first two indices, hence there are no ambiguities
arising from rewriting [t_ij,t_ik+t_jk] = [t_ji,t_jk+t_ik] by using
the symmetry property t_ij =t_ji.
Furthermore, rewriting [t_ij,t_ik+t_jk] = [t_ij,t_(ij)k] = [t_ij,t_k(ij)]
by using the identities (<ref>), one finds 
from the pseudo-naturality structure of t_k(ij) in the right entry
a modification which agrees with Γ_ijk.



For the complete description of a categorified version
of the infinitesimal braid relations, one would also 
require suitable coherence conditions for the modifications Γ_ijk.
One could then attempt to weaken the strict identities (1) and (2) 
from Proposition <ref> to identities which hold true up
to suitable modifications, playing the role of pentagonators and hexagonators
for a weakened variant of the semi-strict braided monoidal category objects from 
Definition <ref>. To the best of our knowledge, this 
is relatively unexplored territory, with the notable exception given by 
exploratory works of Cirio and Faria Martins <cit.>.
In our present paper, we will not attempt to address these coherence issues.



For our infinitesimal 2-braidings t in Theorem <ref>,
which are constructed from 2-shifted Poisson structures π on the CDGA A, we can use
the formulas from Remark <ref> in order to provide 
a more explicit basis expression for the modifications Γ_ijk
in Proposition <ref>. Denoting M=M_i, N=M_j and L=M_k,
we then find for the bottom horizontal arrow in (<ref>)

    t_(M⊗_A N,L),(M⊗_A N,L) (t_M,N⊗𝕀)(⊗w_i⊗u_q⊗v_s)

        =  -(-1)^|w_j| (|L_s^ t|-1) 
⟨π^(2),
s^-1_((t_M,N)_iq^  jr)⊗_A s^-1_(L_s^ t)
⟩⊗w_j⊗u_r⊗v_t  ,

where the components (t_M,N)_iq^  jr are defined in 
(<ref>) and 
the second term in (<ref>) vanishes because k=𝕀:L→ L
is the identity in the present case.


To conclude this subsection, we carry out briefly an explicit study of the 
identities (1) and (2) in Proposition <ref>
modulo ħ^3 by taking as ansatz the traditional Drinfeld associator

Φ(𝔞,𝔟) = 1 + ħ^224 [𝔞,𝔟]  mod ħ^3  ,

see e.g. <cit.>. Using the 
identity (<ref>), one immediately checks 
that the identity (<ref>) holds true strictly modulo ħ^3, 
i.e. without the need for any modifications.
In contrast to this, the identities (<ref>) do not hold true strictly modulo ħ^3.
Concretely, bringing all terms in (<ref>) to the left-hand side, one finds that

ħ^224([t_12, t_13+ t_23] - [t_13,t_12+ t_32])
  ħ^224(Γ_123-Γ_132)  0
 
only holds true weakly up to a certain linear combination of the _[[ħ]]^[-1,0]-enriched modifications
from Proposition <ref>. Similarly, bringing all terms in (<ref>) to the left-hand side,
one finds that

ħ^224([t_23, t_21+ t_31] - [t_13,t_12+ t_32])
  ħ^224(Γ_231-Γ_132)  0

only holds true weakly up to a certain linear combination of the _[[ħ]]^[-1,0]-enriched modifications
from Proposition <ref>. This illustrates that the strict identities
from Proposition <ref>, and the associated concept of semi-strict
braided monoidal category objects from Definition <ref>, is highly restrictive
even at low orders in the deformation parameter ħ and hence they should be relaxed to weaker concepts.






§ EXAMPLES


The aim of this section is to study explicit examples of our 
infinitesimal 2-braidings from Theorem <ref>.
The examples we consider are based on L_∞-algebras,
which are a generalization of differential graded Lie algebras
for which the Jacobi identities hold true only up to homotopy coherence data.
More precisely, an L_∞-algebra consists of -graded vector space L^♯,
which we always assume to be of finite total dimension (L^♯)=∑_i∈(L^i)<∞,
together with a family of brackets ℓ_n : L^♯⊗ n→ L^♯
of degree 2-n, for all n≥ 1, which satisfy the homotopy Jacobi identities.
In particular, the unary bracket ℓ_1 defines a differential on L^♯,
the binary bracket ℓ_2 defines a Lie bracket on the cochain complex
(L^♯,ℓ_1), and the ternary bracket ℓ_3
provides a homotopy witnessing the Jacobi identity for ℓ_2.
See e.g. <cit.> for the complete definition of the concept of
L_∞-algebras and an excellent introduction to their basic theory.


Associated to an L_∞-algebra L= (L^♯,{ℓ_n}) 
is its Chevalley-Eilenberg CDGA ^∙(L). The underlying 
-graded commutative algebra of ^∙(L) is given by

^♯(L) := (L^♯∗[-1])

and the differential _ :=∑_n≥ 1ℓ_n^∗ [-1] is defined on the generators L^♯∗[-1] by dualizing and shifting
the brackets ℓ_n : L^♯⊗ n→ L^♯.
The Chevalley-Eilenberg CDGA ^∙(L) of an L_∞-algebra
of finite total dimension is finitely generated and semi-free, hence it satisfies
our hypotheses from Section <ref>.
The Chevalley-Eilenberg CDGA ^∙(L) admits a geometric interpretation as the CDGA of functions
on the formal quotient stack BL:=[pt/L], which is an infinitesimal
analogue of the classifying stack BG:=[pt/G] of a group G.


The symmetric monoidal dg-category _^∙(L)^fgsf
of finitely generated semi-free ^∙(L)-dg-modules
describes representations of the L_∞-algebra L
and their ∞-morphisms.
Our result in Theorem <ref> then implies that,
given any 2-shifted Poisson structure on the CDGA ^∙(L),
one obtains a first-order deformation of the homotopy 2-category
𝖧𝗈_2(_^∙(L)^fgsf) 
of the representation category of L into a braided monoidal category.
Drawing analogies with the standard deformation theory 
of the (strict) representation category 𝐑𝐞𝐩() of an ordinary Lie algebra  
and its relationship to the quantum group U_q, see
e.g. <cit.> and <cit.>, one may interpret 
our deformation constructions as a first step towards an
L_∞-algebraic generalization of quantum groups.



 §.§ Ordinary Lie algebras


Let us start with the simplest
example which is given by the case where the L_∞-algebra L
is an ordinary (finite-dimensional) Lie algebra =(,[ · , · ]). This means that 
the underlying -graded vector space is , concentrated in degree 0,
and all brackets but the Lie bracket ℓ_2:=[ · , · ] : ⊗→ are trivial.
To simplify our calculations below, we shall choose a basis {x_a∈:a=1,…,()}
and denote the structure constants by [x_a,x_b]=f^c_ab x_c. The Chevalley-Eilenberg
CDGA ^∙() is then given by the -graded commutative algebra


^♯() = [{θ^a}]

which is freely generated by ()-many generators θ^a
of degree 1, for a=1,…,(), together with the differential

_(θ^a) = -12  f^a_bc θ^b θ^c


which is determined by the structure constants. Here and in what follows we 
shall always use the standard summation convention according to which summations over repeated indices are suppressed.
Note that the square-zero condition _ _(θ^a)=0 follows from the Jacobi identity
of the Lie bracket after expressing the latter in terms of the structure constants.


Since the input datum for Theorem <ref> is a 
2-shifted Poisson structure on ^∙() in the sense
of Definition <ref>, we first have to 
understand and characterize the possible choices for such data.
For the ^∙()-dg-module of derivations 𝖳_^∙()
we choose the basis {∂_a∈𝖳_^∙() : a=1,…,()}
which is defined by ∂_a(θ^b)=δ_a^b, for all a,b. Note that
each basis element ∂_a has degree -1 since all θ^b are of degree 1.
The underlying -graded commutative algebra of the completed 2-shifted polyvectors from Definition
<ref> is then given by

(^∙(),2)^♯ = [{θ^a}][[{s^3 ∂_a}]]  ,

where [[ · ]] denotes formal power series. Note that the generators 
s^3∂_a are of degree 3-1=2 in (^∙(),2).

Any 2-shifted Poisson structure π=∑_m≥ 2π^(m)∈(^∙(),2)^4 
on ^∙() is of the form


π = π^(2) = 12 π^ab  s^3∂_a  s^3∂_b  , 

where π^ab∈  are coefficients which
must satisfy the -invariance property

f_cd^a  π^d b +f_cd^b π^a d = 0  .




Using that θ^a is of degree 1 and that s^3∂_a is of degree 2, 
one finds by a simple degree-counting argument that (<ref>) is
the only possible form for a completed 2-shifted polyvector of degree 4 and weight ≥ 2.
The Maurer-Cartan equation from Definition <ref> for such π,
see also Remark <ref>, is equivalent to the two individual equations
_(π)=0 and {π,π}=0. The second equation holds true automatically
because (<ref>) has constant coefficients, while the first equation
is equivalent to the -invariance property (<ref>).
To see the latter point explicitly, one uses the Leibniz rule
for the differential _ and 
_(s^3∂_a) = -f_ab^c θ^b s^3∂_c.



Note that the 2-shifted Poisson structures π from 
Proposition <ref> 
are canonically identified with 
invariant symmetric tensors π =12 π^ab x_a x_b∈ (^2)^.
Hence, our results are compatible with the results of Safronov <cit.>
who has computed the 2-shifted (and also the 1-shifted) Poisson
structures on the classifying stack BG = [pt/G] of an algebraic group G.


Consider now the homotopy 2-category
_^∙()= 𝖧𝗈_2(_^∙()^fgsf)
of the symmetric monoidal dg-category of finitely generated semi-free ^∙()-dg-modules,
which is the object of focus in our deformation construction from Theorem <ref>.
Let us first illustrate the sense in which this category describes 
weak representations of the Lie algebra . By definition, given any object M∈_^∙(),
one has a -graded vector space of generators W^♯ such that M^♯≅^♯()⊗ W^♯.
Choosing any basis {w_i∈ W^♯} for the generators, the differential on M
is completely characterized by specifying _M(⊗ w_i)∈ M, for all i,
which when expanded in terms of the chosen bases reads as

_M(⊗w_i) =  M_i^ j⊗w_j  = ∑_n=0^∞1n! M_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n⊗w_j  ,

where M_a_1… a_n i^a_1… a_n ij∈ are coefficients. 
The square-zero condition _M_M(⊗ w_i)=0 gives a tower of compatibility conditions
between the coefficients M_a_1… a_n i^a_1… a_n ij and the structure constants f^a_bc 
of the Lie algebra , which express a concept of Lie algebra representation up to coherent homotopies.
Strict representations of  on a cochain complex W are recovered in the special case
where _M(⊗ w_i) = M_i^ j⊗ w_j + M_ai^  j θ^a ⊗ w_j
is of polynomial degree ≤ 1 in θ^a. The first term in this expression defines
a differential _W on W^♯ and the second term defines a strict Lie algebra action map 
⊗ W → W which is a cochain map for the differential of W = (W^♯,_W).


In addition to weak representations of the Lie algebra , the homotopy 2-category
_^∙()= 𝖧𝗈_2(_^∙()^fgsf)
also describes a concept of ∞-morphisms
between such weak representations, i.e. weakly -equivariant maps.
(The 2-morphisms in _^∙() are homotopies between such ∞-morphisms.)
Indeed, a morphism h : M→ M^' in _^∙() is by definition
a cochain map which commutes with the left ^∙()-actions.
Hence, by choosing a basis {w_i∈ W^♯} for M^♯≅^♯()⊗ W^♯ 
and a basis {w_i^'∈ W^'♯} for M^'♯≅^♯()⊗ W^'♯,
we can expand 

h(⊗w_i) = h_i^ j⊗w_j^' =  ∑_n=0^∞1n! h_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n⊗w_j^'  ,

where h_a_1… a_n i^a_1… a_n ij∈ are coefficients.
The cochain map condition _M^'(h(⊗ w_i)) = h(_M(⊗ w_i)) 
gives a tower of compatibility conditions
between the coefficients h_a_1… a_n i^a_1… a_n ij 
and the structure constants f^a_bc of the Lie algebra , 
which express a concept of -equivariance up to coherent homotopies.
Strict morphisms h : M→ M^' between strict -representations 
on cochain complexes W and W^' are recovered in the special case
where h(⊗ w_i) = h_i^ j⊗ w^'_j is constant in
θ^a. 


The following result is a direct consequence of Theorem <ref>
and Proposition <ref>.

Every invariant symmetric tensor 
π =12 π^ab x_a x_b∈ (^2)^
defines via Proposition <ref> 
and Theorem <ref> a γ-equivariant infinitesimal 
2-braiding t on the homotopy 2-category _^∙() of the
symmetric monoidal dg-category of finitely generated semi-free ^∙()-dg-modules.



Let us recall that explicit formulas for the components of the 
γ-equivariant infinitesimal 2-braiding t 
are given in (<ref>). Note that the 
morphism components t_M,N in (<ref>)
involve the de Rham differential

(_⊗𝕀)(_M(⊗w_i))    =  
∑_n=1^∞1(n-1)! M_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n-1_(θ^a_n)⊗w_j

of (<ref>), as well as (_⊗𝕀)(_N(⊗ u_q)).
Hence, the components t_M,N in (<ref>) are in general 
non-constant polynomials in θ^a, unless both M and N are strict -representations.
Similar, the homotopy components t_(M,N),(M^',N^')(h⊗k) 
in (<ref>) involve the de Rham differential

(_⊗𝕀)(h(⊗w_i))    =  
∑_n=1^∞1(n-1)! h_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n-1_(θ^a_n)⊗w^'_j

of (<ref>), as well as
(_⊗𝕀)(k(⊗ u_q)).
Hence, the homotopies t_(M,N),(M^',N^')(h⊗k)  
in (<ref>) are in general non-trivial,
unless both h and k are strict morphisms.


As a direct consequence of Proposition <ref>
and Proposition <ref>, we obtain the following result.

Every invariant symmetric tensor 
π =12 π^ab x_a x_b∈ (^2)^
defines via Corollary <ref> and
Proposition <ref>
a braided monoidal deformation to all orders in ħ
of the full subcategory of _^∙()∈_[[ħ]]^[-1,0],ps 
which consists of all ^∙()-dg-modules corresponding to strict -representations.


Combining our observations from
Example <ref> and Remark <ref>,
we obtain that the modifications Γ_ijk in Proposition <ref>
are all trivial for strict -representations because in this case
(t_M,N)_iq^  jr are constant polynomials in θ^a, hence
_((t_M,N)_iq^  jr)=0. This implies that
the hypotheses of Proposition <ref> are satisfied,
which yields the desired braided monoidal deformation to all orders in ħ.



Our result in Corollary <ref> is a homotopical
generalization of the deformation constructions in
<cit.> and <cit.>. We would like to stress that our generalization 
is non-trivial because, even though we restrict ourselves to strict -representations,
the morphisms in the full subcategory from Corollary <ref>
are general (i.e. non-strict) ∞-morphisms between strict -representations. 
Recalling our observations from Remark <ref>, this implies in particular
that the infinitesimal braiding t is a genuine pseudo-natural transformation,
i.e. it has non-trivial homotopy components t_(M,N),(M^',N^')(h⊗k),
and so are the associator and braiding of the deformed category from
Proposition <ref> and Corollary <ref>.





 §.§ String Lie 2-algebras


String Lie 2-algebras are simple examples of L_∞-algebras
which arise as shifted central extensions 
determined by an ordinary (finite-dimensional) Lie algebra =(,[ · , · ]), concentrated
in degree 0, and a 3-cocycle κ:⋀^3→ on .
Such L_∞-algebras appeared first in <cit.>, 
however the name “string Lie 2-algebras” has only been established later.
The resulting L_∞-algebra _κ
is given by the -graded vector space defined by _κ^0:=,
_κ^-1:= and _κ^i:=0, for all i∈∖{-1,0},
and the non-trivial brackets read as

ℓ_2(x,y)  := [x,y]  ,    ℓ_3(x,y,z) :=  κ(x,y,z)  ,

for all x,y,z∈. Let us choose again a basis {x_a∈} and denote the structure constants
by [x_a,x_b]=f^c_ab x_c and κ(x_a,x_b,x_c) = κ_abc.


The Chevalley-Eilenberg CDGA ^∙(_κ)
of the string Lie 2-algebra _κ
is then given by the -graded commutative algebra


^♯(_κ) = [{θ^a},ν]

which is freely generated by ()-many generators θ^a
of degree 1, for a=1,…,(), and a single generator ν
of degree 2, together with the differential

_(θ^a) = -12  f^a_bc θ^b θ^c  ,    _(ν) = -13! κ_abc θ^a θ^b θ^c


which is determined by the structure constants. 
Note that the square-zero condition _ _(θ^a)=0 follows from the Jacobi identity
of the Lie bracket and _ _(ν)=0 
follows from the 3-cocycle condition of κ.


Let us now characterize the
2-shifted Poisson structures on ^∙(_κ) in the sense
of Definition <ref>.
For the ^∙(_κ)-dg-module of derivations 𝖳_^∙(_κ)
we choose the basis {{∂_a},∂_ν}
which is defined by ∂_a(θ^b)=δ_a^b, ∂_a(ν)=0,
∂_ν (θ^a)=0 and ∂_ν(ν)=1, for all a,b. Note that
each basis element ∂_a has degree -1 and ∂_ν has degree -2.
The underlying -graded commutative algebra of the completed 2-shifted polyvectors from Definition
<ref> is then given by

(^∙(_κ),2)^♯ = [{θ^a},ν][[{s^3 ∂_a},s^3∂_ν]]  .

where [[ · ]] denotes formal power series. Note that the generators 
s^3∂_a are of degree 3-1=2 
and that s^3∂_ν is of degree 3-2=1 
in (^∙(_κ),2).

Any 2-shifted Poisson structure 
π=∑_m≥ 2π^(m)∈(^∙(_κ),2)^4 
on ^∙(_κ) is of the form


π = π^(2) =  12 π^ab  s^3∂_a  s^3∂_b + π̃_b^a θ^b s^3∂_a s^3∂_ν  , 

where π^ab∈ and π̃_b^a∈ are coefficients
which must satisfy the properties

f_cd^a π^db + f_cd^b π^ad    = 0  , 

f_bc^d π̃^a_d - f_bd^a π̃^d_c + f_cd^a π̃^d_b -κ_bcd π^da     = 0  ,


π^ac π̃_c^b + π^bc π̃_c^a     =0  . 




The general form (<ref>) for a 
completed 2-shifted polyvector of degree 4 and weight ≥ 2
follows from a simple degree counting argument. (Note that 
products of the form s^3∂_ν s^3∂_ν
vanish in the CDGA (^∙(_κ),2) 
because s^3∂_ν has degree 1, hence s^3∂_ν s^3∂_ν = 
-s^3∂_ν s^3∂_ν =0.)
The Maurer-Cartan equation from Definition <ref> for such π,
see also Remark <ref>, is equivalent to the two individual equations
_(π)=0 and {π,π}=0. 
The first equation _(π)=0 can be evaluated using
the Leibniz rule for the differential _ and

_(s^3∂_a) = -f_ab^c θ^b s^3∂_c +
12 κ_abc θ^b θ^c s^3∂_ν  ,    _(s^3∂_ν) = 0  .

From this one finds the first two properties (<ref>) and (<ref>).
The second equation {π,π}=0 can be evaluated
using the definition (<ref>) and properties (<ref>)
of the Schouten-Nijenhuis bracket, which leads to the third property (<ref>).



As a consequence of (<ref>), the first term in (<ref>) is equivalent to the
datum of an invariant symmetric tensor π =12 π^ab x_a x_b∈(^2)^
on the underlying Lie algebra  of the string Lie 2-algebra _κ.
The second term in (<ref>) is equivalent to a linear map 
π̃:→ , x_a↦π̃_a^b x_b
on the underlying Lie algebra. Property 
(<ref>) then demands that the linear map π̃ annihilates
the invariant symmetric tensor π according to


(π̃⊗𝕀_)(π) + (𝕀_⊗π̃)(π) = 0  .

The remaining property (<ref>) can be written equivalently as
 
π̃([x,y]) =  [π̃(x),y] + [x,π̃(y)] + 
(κ(x,y,-)⊗𝕀_)(π) + (𝕀_⊗κ(x,y,-))(π)    ,


for all x,y∈, which means that the linear map π̃:→ 
is a derivation of the underlying Lie algebra , up to corrections arising from
the 3-cocycle κ:⋀^3→ and the invariant symmetric tensor π∈(^2)^.


Our 2-shifted Poisson structures on ^∙(_κ)
are similar, but as it seems not identical, to the quasi-invariant symmetric
tensors on crossed modules of Lie algebras from <cit.>.
A direct comparison between these two structures is difficult because 
our preferred model for the string Lie 2-algebra _κ is given by
a finite-dimensional L_∞-algebra with a non-trivial ℓ_3 bracket (<ref>), 
while <cit.> use a different model which is given by an
infinite-dimensional crossed module of Lie algebras 
that encodes the cocycle κ less directly.
Our suspicion that these two concepts are slightly different
will be strengthened by the following example.



In order to illustrate Proposition <ref>,
let us consider the example where = 𝔰𝔩_2 
is the special linear Lie algebra of order 2. We work with
the usual basis {x_+ ,x_- ,x_3∈𝔰𝔩_2} 
whose Lie bracket relations are

[x_+,x_-]  = x_3  ,    [x_3,x_±] = ±2 x_±  .

Using the rescaled Killing form 
( · , · ):𝔰𝔩_2⊗𝔰𝔩_2→
whose non-vanishing values are given by (x_3,x_3) =1 and (x_+,x_-)=(x_-,x_+)=12,
we define the 3-cocycle κ(x,y,z):= (x,[y,z]). In terms of our basis,
this 3-cocycle is given by

κ(x_+,x_-,x_3) = 1

and total antisymmetry in its arguments.
This defines the string Lie 2-algebra (𝔰𝔩_2)_κ.


Since 𝔰𝔩_2 is a simple Lie algebra, there exists a unique (up to scale)
invariant symmetric tensor, which is concretely given by

π =  λ ( x_+   x_- +14 x_3^2)  ∈  (^2 𝔰𝔩_2)^𝔰𝔩_2  ,

where λ∈ is any parameter. 
Assuming for the moment that λ≠ 0 is non-zero, the general solution
for the linear map π̃ : 𝔰𝔩_2→𝔰𝔩_2
to the equation (<ref>), or 
the equivalent equation (<ref>), is given by

π̃(x_+) = a x_+ + b x_3  ,  π̃(x_-) = -a x_- + c x_3  ,  π̃(x_3) = -2 c x_+ -2 b  x_-  ,

where a,b,c∈ are arbitrary parameters. Evaluating the equivalent form (<ref>)
of the remaining equation (<ref>) on x=x_+ and y=x_-, one finds 
that it does not admit any solutions for λ≠ 0. This implies that, 
for every 2-shifted Poisson structure on ^∙((𝔰𝔩_2)_κ) as in
Proposition <ref>, the 
invariant symmetric tensor (<ref>) must necessarily be trivial π=0.
In this case the equations in (<ref>),
see also (<ref>) for their equivalent forms, simplify to the single condition
that π̃:𝔰𝔩_2→𝔰𝔩_2 is a Lie algebra derivation,
i.e. π̃([x,y]) = [π̃(x),y] + [x,π̃(y)],
for all x,y∈𝔰𝔩_2. Hence, the 2-shifted Poisson structures on ^∙((𝔰𝔩_2)_κ)
are classified by the Lie algebra derivations π̃ on 𝔰𝔩_2.
This behavior is drastically different to the case of ordinary Lie algebras  
from Proposition <ref> and the
quasi-invariant symmetric tensors on crossed modules of Lie algebras
from <cit.> and <cit.>.


The following result is a direct consequence of Theorem <ref>,
Proposition <ref> and Remark <ref>.

Every pair (π,π̃) consisting of a 
invariant symmetric tensor π =12 π^ab x_a x_b∈ (^2)^
and a linear map π̃ : → , x_a↦π̃_a^b x_b
which satisfies the properties in (<ref>)
defines via Proposition <ref>
and Theorem <ref> a γ-equivariant infinitesimal 
2-braiding t on the homotopy 2-category _^∙(_κ) of the
symmetric monoidal dg-category of finitely generated semi-free ^∙(_κ)-dg-modules.



Recalling the explicit formulas for the components of the 
γ-equivariant infinitesimal 2-braiding t 
from (<ref>), one immediately observes 
by using also (<ref>) that the morphism components t_M,N 
are in general non-constant polynomials in θ^a 
in the case where π̃^a_b≠ 0 is non-trivial. 
(As illustrated in Example <ref>, this seems to be the most relevant case.)
Hence, the modifications Γ_ijk from Proposition <ref> and Example
<ref> are in general non-trivial. This even holds true for
representation of the string Lie 2-algebra _κ which are strict
in the sense that the expansions as in (<ref>) 
of the differential _M are polynomials in θ^a and ν of degree ≤ 1.
As a consequence, we currently do not have a meaningful generalization 
of the all-order deformation result from Corollary <ref>
to the case of ^∙(_κ). Hence, further work as outlined in
Remark <ref> is needed in order to extend
our result in Corollary <ref> to a braided monoidal
deformation up to all orders in ħ.







§ ACKNOWLEDGMENTS

We would like to thank Jon Pridham for useful discussions.
C.K. is supported by an EPSRC doctoral studentship. 
A.S. gratefully acknowledges the support of 
the Royal Society (UK) through a Royal Society University 
Research Fellowship (URF\R\211015)
and Enhancement Grants (RF\ERE\210053 and 
RF\ERE\231077).










§ HIGHER-CATEGORICAL FRAMEWORK


In this appendix, we introduce a suitable 
higher-categorical framework in which the deformation constructions
from Section <ref> take place. 
The basic objects of our interest are categories 
which are enriched over the symmetric monoidal category _R^[-1,0] 
of 2-term cochain complexes (concentrated in degrees -1 and 0) 
of modules over a commutative -algebra R. 
We will show below that these objects assemble
into a suitably weakened kind of 3-category _R^[-1,0],ps whose 1-morphisms
are _R^[-1,0]-enriched functors, 2-morphisms are 
_R^[-1,0]-enriched pseudo-natural transformations and 
3-morphisms are _R^[-1,0]-enriched modifications. This 3-category carries a symmetric monoidal
structure, which allows one to define a concept of _R^[-1,0]-enriched
braided monoidal categories whose associators, unitors and braidings are 
enriched pseudo-natural transformations.
This increased flexibility arising from pseudo-naturality is essential 
for our construction of infinitesimal 2-braidings in Section <ref>.
We assume that the reader has some familiarity with 
enriched category theory, see e.g. <cit.> for a comprehensive treatment, 
or <cit.> and <cit.> for more concise introductions.




 §.§ 3-categories with pseudofunctorial composition


Let us recall that a strict 3-category can be defined as a category which is
enriched over the category Cat^str of strict 2-categories and strict 2-functors.
We require a slightly weakened variant of this concept where the enrichment
category is generalized to the category Cat^ps of strict 2-categories
and pseudo-functors. See e.g. <cit.> for a detailed
description of this category and, in particular, of its strictly associative and unital composition.
The category Cat^ps admits finite products and a terminal object,
given by the 2-category 1∈Cat^ps consisting of a single 0-cell
and its identity 1-cell and 2-cell, hence it is symmetric monoidal with respect to the 
Cartesian symmetric monoidal structure.

We denote by 

Cat := Cat^ps
the 2-category whose


  * objects are Cat^ps-enriched categories,

  * 1-morphisms are Cat^ps-enriched functors, and

  * 2-morphisms are Cat^ps-enriched natural transformations.



To illustrate how an object ∈Cat admits an interpretation
in terms of a weakened kind of 3-category, let us spell out, in some detail, 
the data defining . By definition of an enriched category,  has a class 
of objects and, for each pair of objects c,c^'∈, a strict 2-category
(c,c^')∈Cat^ps of morphisms. Since the 2-category
(c,c^') consists of 0-cells, 1-cells and 2-cells, there are three layers
of morphisms in ∈Cat, as required for a 3-category. The enriched category 
comes further with identity morphisms 𝕀_c : 1→(c,c), or equivalently
0-cells 𝕀_c∈(c,c), and composition morphisms

∗ :  (c^',c^'')×(c,c^')  ⟶ (c,c^'')

in Cat^ps, which in our case are pseudo-functors
between strict 2-categories. The compositions ∗ must be strictly associative 
and strictly unital with respect to the identities 𝕀_c.
Note that the relaxation to pseudo-functors makes our variant of 3-categories 
from Definition <ref> more general than strict 3-categories, 
for which the analogous composition morphisms ∗ must be strict 2-functors.


The 2-category Cat carries a symmetric monoidal structure
which is determined by the Cartesian symmetric monoidal structure of the enrichment category Cat^ps,
see e.g. <cit.> for the details. The monoidal product ×∈Cat
of ,∈Cat is the Cat^ps-enriched category
whose objects are pairs (c,d)∈× of objects c∈ and d∈
and whose morphisms (×)((c,d),(c^',d^')):=(c,c^')×(d,d^')
∈Cat^ps are given by the Cartesian symmetric monoidal structure on Cat^ps.
The monoidal unit ℐ∈Cat is the Cat^ps-enriched category
with a single object, say ⋆∈ℐ, and morphisms ℐ(⋆,⋆):=
1∈Cat^ps the terminal object. The symmetric braiding


flip_,^ : × ⟶ ×
is the Cat^ps-enriched functor which acts on objects by flipping 
(c,d)↦ (d,c) and on morphisms through the symmetric braiding on Cat^ps

(×)((c,d),(c^',d^')) =  (c,c^')×(d,d^') 
≅⟶  (d,d^')×(c,c^') =  (×)((d,c),(d^',c^'))  .


Making use of this symmetric monoidal structure on Cat,
one can define a concept of symmetric monoidal category objects in Cat.

A symmetric monoidal category object in Cat is a tuple 
(,⊠,u,α,λ,ρ,τ)
consisting of the following data:


  (i) an object ∈Cat,

  (ii) a 1-morphism ⊠ : ×→ in Cat (called monoidal product),

  (iii) a 1-morphism u : ℐ→ in Cat (called monoidal unit),

  (iv) invertible 2-morphisms

    𝒞×𝒞×𝒞      𝒞×𝒞
    
    𝒞×𝒞      𝒞["⊠×id_𝒞", from=1-1, to=1-3]
    	["id_𝒞×⊠"', from=1-1, to=3-1]
    	["α"', shorten <=12pt, shorten >=12pt, Rightarrow, from=1-3, to=3-1]
    	["⊠", from=1-3, to=3-3]
    	["⊠"', from=3-1, to=3-3]
           𝒞
    ℐ×𝒞      𝒞×ℐ
       𝒞×𝒞["≅"', from=1-2, to=2-1]
    	["≅", from=1-2, to=2-3]
    	["u×id_𝒞"', from=2-1, to=3-2]
    	["id_𝒞× u", from=2-3, to=3-2]
    	[""name=0, anchor=center, inner sep=0, "⊠"description, from=3-2, to=1-2]
    	["λ", shorten <=5pt, shorten >=9pt, Rightarrow, from=2-1, to=0]
    	["ρ"', shorten <=5pt, shorten >=9pt, Rightarrow, from=2-3, to=0]

in Cat (called associator, left unitor and right unitor),


  (v) an invertible 2-morphism 

    𝒞×𝒞      𝒞
       𝒞×𝒞[""name=0, anchor=center, inner sep=0, "⊠", from=1-1, to=1-3]
    	["flip_𝒞,𝒞^"', from=1-1, to=2-2]
    	["⊠"', from=2-2, to=1-3]
    	["τ", shorten <=6pt, shorten >=3pt, Rightarrow, from=0, to=2-2]

in Cat (called symmetric braiding).

These data have to satisfy the usual coherence conditions for symmetric monoidal categories
internal to the 2-category Cat.





 §.§ A symmetric monoidal 3-category of 2-term truncated dg-categories


Let R be a commutative, associative and unital -algebra and consider
the symmetric monoidal category _R of cochain complexes of R-modules
from  Subsection <ref>.
We denote by

_R^[-1,0] ⊆  _R

the full subcategory whose objects are all cochain complexes V∈_R
concentrated in degrees {-1,0}, i.e. V^i=0 for all i∈∖{-1,0}.
We shall often denote such objects by

V  =  (@C=2em
V^-1  [r]^-_V       V^0
)
 ∈  _R^[-1,0]   .

Note that the tensor product ⊗_R on _R does not directly restrict
to the full subcategory _R^[-1,0]⊆_R because
V⊗_R W∈_R is, in general, concentrated in degrees {-2,-1,0},
for V,W∈_R^[-1,0]. This issue can be resolved by composing ⊗_R with
the good truncation functor τ^[-1,0] : _R →_R^[-1,0] to define
the truncated tensor product


⊗_R : 
@C=3em
_R^[-1,0]×_R^[-1,0][r]^-⊆       _R×_R[r]^-⊗_R      
_R [r]^-τ^[-1,0]       _R^[-1,0]
  .

Explicitly, given two objects V,W∈_R^[-1,0], the truncated tensor product reads as

V⊗_R W =  (
@C=4em
(V^-1⊗_R W^0) ⊕(V^0⊗_R W^-1)/_V⊗_R W(V^-1⊗_R W^-1) [r]^-_V⊗_R W      
V^0⊗_R W^0

)  .


One directly checks that _R^[-1,0] forms a symmetric monoidal category
with respect to the truncated tensor product ⊗_R, the monoidal unit R∈_R^[-1,0]
and the symmetric braiding

γ_V,W^ :  V⊗_R W ⟶ W⊗_R V ,  
v⊗_R w ⟼  w⊗_R v  ,

where all Koszul signs are trivial
because there are no odd-odd degree combinations in the truncated tensor product (<ref>).
For later use, let us observe that the symmetric monoidal categories
(_R,⊗_R,R,γ) and (_R^[-1,0],⊗_R,R,γ)
are related by a lax symmetric monoidal functor.

The truncation functor τ^[-1,0] : _R →_R^[-1,0] is canonically lax symmetric monoidal.


The functor τ^[-1,0] preserves the monoidal units strictly, i.e. τ^[-1,0](R)=R,
and the lax structure τ^[-1,0](V) ⊗_R τ^[-1,0](W)→τ^[-1,0](V⊗_R W) for the tensor product of V,W∈_R 
is given by the obvious cochain map (drawn vertically)


@C=4em@R=2em
[d] (V^-1_V V^-2⊗_R 𝖹^0(W)) ⊕(𝖹^0(V)⊗_R W^-1_W W^-2)/_V⊗_R W (V^-1⊗_R W^-1) [r]^-_V⊗_R W       𝖹^0(V)⊗_R 𝖹^0(W)
[d]

(V⊗_R W)^-1/_V⊗_R W(V⊗_R W)^-2[r]_-_V⊗_R W     𝖹^0(V⊗_R W)

  ,

where 𝖹^0(U) := (_U : U^0 → U^1) denotes the submodule of 0-cocycles
associated with a cochain complex U∈_R.


The goal of the remaining part of this subsection is to provide details
for the following result.

There exists a 3-category

_R^[-1,0],ps  ∈ Cat

in the sense of Definition <ref> whose


  * objects are _R^[-1,0]-enriched categories,

  * 1-morphisms are _R^[-1,0]-enriched functors,

  * 2-morphisms are _R^[-1,0]-enriched pseudo-natural transformations, and

  * 3-morphisms are _R^[-1,0]-enriched modifications.

This 3-category admits a symmetric monoidal structure in the sense of
Definition <ref> which agrees on objects, 1-morphisms 
and strictly natural 2-morphisms with the usual one from enriched
category theory, see e.g. <cit.>.


The objects and 1-morphisms in _R^[-1,0],ps are defined 
according to the standard definitions of enriched category theory.
The precise definitions of the 2-morphisms and 3-morphisms are as follows.

Let F,G : → be two _R^[-1,0]-enriched functors between two
_R^[-1,0]-enriched categories  and . A _R^[-1,0]-enriched pseudo-natural
transformation ζ : F⇒ G consists of the following data:


  (i) For each object c∈, an element ζ_c ∈(F(c),G(c))^0 
of degree 0 in the -object (F(c),G(c))∈_R^[-1,0].


  (ii) For each pair of objects c,c^'∈, an R-linear map

ζ_c,c^' :  (c,c^')^0 ⟶ (F(c),G(c^'))^-1  .


These data have to satisfy the following properties:


  (1) For all h∈(c,c^')^0,


G(h)  ζ_c - ζ_c^' F(h) = _( ζ_c,c^'(h))  ,

and for all k∈(c,c^')^-1,

G(k)  ζ_c - ζ_c^' F(k) =  ζ_c,c^'(_(k))  .




  (2) For all c∈,

ζ_c,c(𝕀_c)  = 0  .



  (3) For all h∈(c,c^')^0 and h^'∈(c^',c^'')^0,

ζ_c,c^''(h^' h) =  ζ_c^',c^''(h^')  F(h) + G(h^') ζ_c,c^'(h)  .





Note that in the special case where ζ_c,c^'=0, for all c,c^'∈, 
Definition <ref> reduces to the usual definition of a _R^[-1,0]-enriched (strict) 
natural transformation.



Let ζ,η : F⇒ G: → be two 
_R^[-1,0]-enriched pseudo-natural transformations.
A _R^[-1,0]-enriched modification Γ : ζ⇛η
consists of the following data:


  (i) For each object c∈, an element Γ_c ∈(F(c),G(c))^-1 
of degree -1 in the -object (F(c),G(c))∈_R^[-1,0].

These data have to satisfy the following properties:


  (1) For all c∈, 

ζ_c-η_c = _(Γ_c)  .



  (2) For all h∈(c,c^')^0,

Γ_c^'  F(h) + ζ_c,c^'(h) = η_c,c^'(h)+ G(h) Γ_c  .




In the following paragraphs we describe explicitly
the symmetric monoidal 3-category structure of _R^[-1,0],ps 
in the sense of Definitions <ref> and <ref>.
To avoid overloading this appendix, we shall only spell out the relevant data 
but not the verification of their properties,  which are mostly straightforward calculations.



  
The homomorphism 2-categories: Given two _R^[-1,0]-enriched
categories  and , we denote by _R^[-1,0],ps(,)∈Cat^ps
the strict 2-category whose objects are _R^[-1,0]-enriched functors,
1-morphisms are _R^[-1,0]-enriched pseudo-natural transformations and
2-morphisms are _R^[-1,0]-enriched modifications. The composition
of two 1-morphisms


    𝐂      𝐃[""name=0, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"',curve=height=32pt, from=1-1, to=1-3]
    	["ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=0]
    	["η"', shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=2]
      ∘⟼  𝐂      𝐃[""name=1, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"', curve=height=32pt, from=1-1, to=1-3]
    	["η∘ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=2]

in the 2-category _R^[-1,0],ps(,) is defined by

(η∘ζ)_c    :=  η_c ζ_c  ,

(η∘ζ)_c,c^'(h)    :=  η_c,c^'(h) ζ_c + η_c^' ζ_c,c^'(h)  ,


for all c,c^'∈ and h∈(c,c^')^0. This composition
is strictly associative and strictly unital with respect to the identity 1-morphisms
_F : F⇒ F : → defined by (_F)_c = 𝕀_F(c) and
(_F)_c,c^'=0.


The horizontal composition of two 2-morphisms


    𝐂      𝐃[""name=0, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "H"', curve=height=36pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-3]
    	[""name=3, anchor=center, inner sep=0, " ζ^'", curve=height=-16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=0]
    	[""name=4, anchor=center, inner sep=0, "ζ "', curve=height=16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=0]
    	[""name=5, anchor=center, inner sep=0, "η "', curve=height=16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	[""name=6, anchor=center, inner sep=0, " η^'", curve=height=-16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=3]
    	["Θ", shorten <=5pt, shorten >=5pt, triple, from=5, to=6]
      ∘⟼  𝐂         𝐃[""name=1, anchor=center, inner sep=0, "H"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=3, anchor=center, inner sep=0, " η^'∘ζ^'", curve=height=-20pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=1]
    	[""name=4, anchor=center, inner sep=0, "η∘ζ "', curve=height=20pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=1]
    	["Θ∘Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=3]

in the 2-category _R^[-1,0],ps(,) is defined by

(Θ∘Γ)_c :=  Θ_c ζ_c + η^'_c Γ_c  ,


for all c∈. This composition is strictly associative and strictly unital
with respect to the identity 2-morphisms
__F : _F⇛_F : F⇒ F : → defined by (__F)_c = 0.


The vertical composition of two 2-morphisms


    𝐂         𝐃[""name=0, anchor=center, inner sep=0, "G"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, " ζ^''", curve=height=-28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=3, anchor=center, inner sep=0, "ζ "', curve=height=28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=4, anchor=center, inner sep=0, "ζ^'"description, shorten <=6pt, shorten >=6pt, Rightarrow, from=1, to=0]
    	["Γ", shorten <=4pt, shorten >=4pt, triple, from=3, to=4]
    	["Γ^'", shorten <=4pt, shorten >=4pt, triple, from=4, to=2]
      ⟼  𝐂         𝐃[""name=0, anchor=center, inner sep=0, "G"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, " ζ^''", curve=height=-28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=3, anchor=center, inner sep=0, "ζ "', curve=height=28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	["Γ^' Γ", shorten <=4pt, shorten >=4pt, triple, from=3, to=2]

in the 2-category _R^[-1,0],ps(,) is defined by

(Γ^' Γ)_c  :=  Γ^'_c + Γ_c  ,


for all c∈. This composition is strictly associative and strictly unital
with respect to the identity 2-morphisms
_ζ : ζ⇛ζ : F⇒ F : → defined by (_ζ)_c = 0.





  
The composition pseudo-functors and identities: Given three _R^[-1,0]-enriched
categories ,  and , we denote by 

∗ : _R^[-1,0],ps(,) ×_R^[-1,0],ps(,)  ⟶ 
_R^[-1,0],ps(,)

the composition pseudo-functor between the homomorphism 2-categories
from the previous paragraph.
On objects in the homomorphism 2-categories,
this pseudo-functor acts in terms of the usual composition of _R^[-1,0]-enriched functors,
which we will denote by

    𝐂   𝐃   𝐄["F", from=1-1, to=1-2]
    	["F^'", from=1-2, to=1-3]
      ∗⟼  𝐂   𝐄["F^'∗ F", from=1-1, to=1-2]
      .

On 1-morphisms in the homomorphism 2-categories, the pseudo-functor ∗ acts as


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=3, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-3, to=1-5]
    	["ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
    	[" ζ^'"', shorten <=6pt, shorten >=6pt, Rightarrow, from=2, to=3]
      ∗⟼  𝐂      𝐄[""name=0, anchor=center, inner sep=0, "F^'∗ F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'∗ G"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ^'∗ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]

with

(ζ^'∗ζ)_c    := ζ^'_G(c) F^'(ζ_c)  ,

(ζ^'∗ζ)_c,c^'(h)    := ζ^'_G(c),G(c^')(G(h)) F^'(ζ_c)
+ζ^'_G(c^') F^'(ζ_c,c^'(h))  ,


for all c,c^'∈ and h∈(c,c^')^0. 
On 2-morphisms in the homomorphism 2-categories, the pseudo-functor ∗ acts as


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=3, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-3, to=1-5]
    	[""name=4, anchor=center, inner sep=0, "ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=5, anchor=center, inner sep=0, " η", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=6, anchor=center, inner sep=0, " ζ^'"', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=2, to=3]
    	[""name=7, anchor=center, inner sep=0, " η^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=2, to=3]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=5]
    	["Γ^'", shorten <=5pt, shorten >=5pt, triple, from=6, to=7]
      ∗⟼  𝐂         𝐄[""name=0, anchor=center, inner sep=0, "F^'∗ F", curve=height=-32pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "G^'∗ G"', curve=height=32pt, from=1-1, to=1-4]
    	[""name=4, anchor=center, inner sep=0, "ζ^'∗ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=5, anchor=center, inner sep=0, " η^'∗η", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ^'∗Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=5]

with

(Γ^'∗Γ)_c := Γ^'_G(c) F^'(ζ_c) + η^'_G(c)  F^'(Γ_c)  ,


for all c∈.


The pseudo-functor coherences for the identities
are trivial, i.e. _F^'∗ F = _F^'∗_F,
but those for the compositions are not. Given any composable diagram of the form


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=3, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=4, anchor=center, inner sep=0, "H^'"', curve=height=32pt, from=1-3, to=1-5]
    	[""name=5, anchor=center, inner sep=0, "G^'"description, from=1-3, to=1-5]
    	["ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	["η"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=2]
    	["ζ^'"', shorten <=5pt, shorten >=5pt, Rightarrow, from=3, to=5]
    	["η^'"', shorten <=5pt, shorten >=5pt, Rightarrow, from=5, to=4]
      ,

the corresponding pseudo-functor coherence

∗_(η^',η),(ζ^',ζ)^ : (η^'∗η)∘(ζ^'∗ζ)      (η^'∘ζ^')∗(η∘ζ)

is given by the _R^[-1,0]-enriched modification
whose components are defined by

(∗_(η^',η),(ζ^',ζ)^)_c := η^'_H(c) ζ^'_G(c),H(c)(η_c) F^'(ζ_c)  ,


for all c∈.


The composition pseudo-functors ∗ are strictly associative and strictly unital
with respect to the identities 𝕀_∈_R^[-1,0],ps(,) 
given by the identity _R^[-1,0]-enriched functors 𝕀_:→.




  
The symmetric monoidal structure: The monoidal product

⊠ :  
_R^[-1,0],ps ×_R^[-1,0],ps
 ⟶ 
_R^[-1,0],ps

is given by the Cat^ps-enriched functor which assigns to a pair
of objects ,∈_R^[-1,0],ps the 
usual _R^[-1,0]-enriched tensor product category ⊠∈_R^[-1,0],ps
and to a pair of 1-morphisms F : →^' and G : →^'
in _R^[-1,0],ps the 
usual _R^[-1,0]-enriched tensor product functor 
F⊠ G : ⊠→^'⊠^'
from <cit.>.  


The monoidal product 
of a pair of 2-morphisms


    𝐂      𝐂^'[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
      𝐃      𝐃^'[""name=0, anchor=center, inner sep=0, "G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-1, to=1-3]
    	["η "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
      ⊠⟼  𝐂⊠𝐃      𝐂^'⊠𝐃^'[""name=0, anchor=center, inner sep=0, "F⊠ G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'⊠ G^'"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ⊠η "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]

in  _R^[-1,0],ps is defined by

(ζ⊠η)_(c,d)    := ζ_c⊗_Rη_d  ,

(ζ⊠η)_(c,d),(c^',d^')(h⊗_R k)    := 
ζ_c,c^'(h)⊗_R G^'(k) η_d + ζ_c^' F(h)⊗_Rη_d,d^'(k)  ,


for all c,c^'∈, d,d^'∈, h∈(c,c^')^0
and k∈(d,d^')^0. Note that for two _R^[-1,0]-enriched 
strictly natural transformations, i.e. ζ_c,c^'=0 and η_d,d^'=0
for all objects c,c^'∈ and d,d^'∈, the tensor
product ζ⊠η agrees with the usual one from enriched category theory.


The monoidal product of a pair of 3-morphisms


    𝐂      𝐂^'[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " ζ^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]
      𝐃      𝐃^'[""name=0, anchor=center, inner sep=0, "G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "η "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " η^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Θ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]
      ⊠⟼  𝐂⊠𝐃         𝐂^'⊠𝐃^'[""name=0, anchor=center, inner sep=0, "F⊠ G", curve=height=-32pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F^'⊠ G^'"', curve=height=32pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, "ζ⊠η "', curve=height=22pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " ζ^'⊠η^'", curve=height=-22pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ⊠Θ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]

in  _R^[-1,0],ps is defined by

(Γ⊠Θ)_(c,d) := Γ_c⊗_R η_d + ζ^'_c ⊗_RΘ_d  ,


for all c∈ and d∈.


The monoidal unit u :ℐ→_R^[-1,0],ps
is the _R^[-1,0]-enriched category

𝐑 ∈   _R^[-1,0],ps

consisting of a single object, say ⋆, with morphisms
𝐑(⋆,⋆):= R∈_R^[-1,0] given by
the monoidal unit of the enrichment category.


The associator and unitors for this monoidal structure 
are canonically given by the ones of the enrichment 
symmetric monoidal category _R^[-1,0].
The component at ,∈_R^[-1,0],ps of the symmetric braiding 


τ_, : ⊠ ⟶ ⊠
is the _R^[-1,0]-enriched functor which sends objects
(c,d)∈⊠ to (d,c)∈⊠
and acts on morphisms via the symmetric braiding of the enrichment 
symmetric monoidal category _R^[-1,0]


@C=5em@R=1.5em
(⊠)((c,d),(c^',d^')) @=[d]@–>[r]     @=[d]
(⊠)((d,c),(d^',c^'))

(c,c^')⊗_R(d,d^') [r]_-γ_(c,c^'),(d,d^')
     (d,d^')⊗_R(c,c^')

  .


Note that the symmetric braiding satisfies the strict hexagon identities, i.e.

τ_⊠,    = (τ_,⊠𝕀_)∗(𝕀_⊠τ_,)  ,

τ_,⊠    = (𝕀_⊠τ_,)∗(τ_,⊠𝕀_)  ,


for all ,,∈_R^[-1,0],ps.
Moreover, the symmetric braiding τ is a Cat^ps-enriched natural 
transformation, which in particular implies that it is strictly natural with respect to 
_R^[-1,0]-enriched functors F : →^', i.e.

(𝕀_⊠F)∗τ_,=τ_^',∗(F⊠𝕀_)  ,

for all ∈_R^[-1,0],ps.


To conclude this subsection, we provide a convenient
definition of braided monoidal category objects
in the symmetric monoidal 3-category _R^[-1,0],ps.
Such objects can be defined with various levels of weakness since
our ambient category _R^[-1,0],ps is a 3-category.
For the purpose of our paper, it suffices to consider 
the following semi-strict variant.

A (semi-strict) braided monoidal category object in _R^[-1,0],ps 
is a tuple (,⊗,1,α,λ,ρ,γ)
consisting of the following data:


  (i) a _R^[-1,0]-enriched category ,


  (ii) a _R^[-1,0]-enriched functor ⊗ : ⊠→ (called monoidal product),


  (iii) a _R^[-1,0]-enriched functor 1 : 𝐑→ (called monoidal unit),


  (iv) invertible _R^[-1,0]-enriched pseudo-natural transformations 
α : ⊗∗ (⊗⊠𝕀_) ⇒⊗∗ (𝕀_⊠⊗)
(called associator), λ : ⊗∗(1⊠𝕀_) ⇒𝕀_ (called left unitor)
and ρ: ⊗∗(𝕀_⊠ 1)⇒𝕀_ (called right unitor),


  (v) an invertible _R^[-1,0]-enriched pseudo-natural transformation
γ : ⊗⇒⊗∗τ_, (called braiding).

These data have to satisfy the usual coherence conditions for braided monoidal categories
internal to the 3-category _R^[-1,0],ps strictly,
i.e. the pentagon, triangle and hexagon diagrams commute strictly 
up to equality of _R^[-1,0]-enriched pseudo-natural transformations.


A (semi-strict) symmetric monoidal category object in _R^[-1,0],ps 
is a braided monoidal category object (,⊗,1,α,λ,ρ,γ)
with the property that the braiding γ squares strictly to the identity,
i.e. the ∘-composition of

@C=3.5em
⊗@=>[r]^-γ      ⊗∗τ_, @=>[r]^-γ∗_τ_,
     ⊗∗τ_,∗τ_,  =  ⊗

is equal to the identity _R^[-1,0]-enriched pseudo-natural transformation _⊗.





 §.§ Homotopy 2-categories of symmetric monoidal dg-categories


Let R be a commutative -algebra and denote by _R the 2-category
of _R-enriched categories (dg-categories), _R-enriched functors (dg-functors)
and _R-enriched strict natural transformations (dg-natural transformations).
We shall regard _R as a 3-category (with trivial 3-morphisms) 
in the sense of Definition <ref>.
The lax symmetric monoidal truncation functor  τ^[-1,0]: _R→_R^[-1,0]
from Lemma <ref> induces a change of base 3-functor

𝖧𝗈_2 :  _R ⟶ _R^[-1,0] ⊆ _R^[-1,0],ps  ,

which sends dg-natural transformations to strict
_R^[-1,0]-enriched pseudo-natural transformations.
Observe that, at the level of objects, this 3-functor assigns to a dg-category ∈_R, 
with objects _0 and morphism cochain complexes (c,c^')∈_R, 
its homotopy 2-category 𝖧𝗈_2()∈_R^[-1,0],ps, 
i.e. the _R^[-1,0]-enriched category with the same objects _0 and morphisms
given by the truncated cochain complexes τ^[-1,0]((c,c^'))∈_R^[-1,0].


Since the enrichment category _R of cochain complexes of R-modules is symmetric monoidal
with respect to the relative tensor product ⊗_R, there exists an induced
symmetric monoidal structure on _R, see e.g. <cit.>.
The 3-functor (<ref>) assigning homotopy 2-categories
is lax symmetric monoidal with the lax structure 
𝖧𝗈_2()⊠𝖧𝗈_2() →𝖧𝗈_2(⊠)
induced from the lax structure of the good truncation functor in
Lemma <ref>, for all ,∈_R.
For later use, we record the following direct consequence of this result.

Let (,⊗,1,γ) be a braided (or symmetric) monoidal dg-category.
Then the homotopy 2-category 𝖧𝗈_2()∈_R^[-1,0],ps
is canonically a braided (or symmetric) monoidal category object in
the symmetric monoidal 3-category _R^[-1,0],ps from Proposition <ref>.



Note that the braided (or symmetric) monoidal category objects in _R^[-1,0],ps 
which are obtained from this construction always have strictly natural
associators, unitors and braidings. Our symmetric monoidal 
3-category _R^[-1,0],ps from Proposition <ref>
is designed to provide more flexibility through the concept 
of enriched pseudo-natural transformations, see also Definition 
<ref>, which will be relevant 
for our deformation constructions in Section <ref>.







10

[BC04]BaezCrans
J. C. Baez and A. S. Crans,
“Higher-dimensional algebra VI: Lie 2-algebras,”
Theory Appl. Categ. 12, 492–538 (2004)
[arXiv:math/0307263 [math.QA]].


[BSZ13]Zhu
C. Bai, Y. Sheng and C. Zhu,
“Lie 2-bialgebras,”
Comm. Math. Phys. 320, 149–172 (2013)
[arXiv:1109.1344 [math-ph]].


[BV15]Voronov
D. Bashkirov and A. A. Voronov,
“r_∞-matrices, triangular L_∞-bialgebras and quantum ∞-groups,”
in: P. Kielanowski, P. Bieliavsky, A. Odzijewicz, M. Schlichenmaier and T. Voronov (eds.),
Geometric Methods in Physics, 
Trends in Mathematics, Birkhäuser, Cham (2015)
[arXiv:1412.2413 [math.QA]].


[CPTVV17]CPTVV
D. Calaque, T. Pantev, B. Toën, M. Vaquié and G. Vezzosi,
“Shifted Poisson structures and deformation quantization,”
J. Topol. 10, no. 2, 483–584 (2017)
[arXiv:1506.03699 [math.AG]].


[Car93]Cartier
P. Cartier,
“Construction combinatoire des invariants de Vassiliev-Kontsevich des nœuds,”
C. R. Acad. Sci. Paris Sér. I Math. 316, no. 11, 1205–1210 (1993).


[CG23]Girelli
H. Chen and F. Girelli,
“Categorical quantum groups and braided monoidal 2-categories,”
arXiv:2304.07398 [math.QA].


[CR08]ChuangRouquier
J. Chuang and R. Rouquier, 
“Derived equivalences for symmetric groups and 𝔰𝔩_2-categorification,” 
Ann. of Math. 167, no. 1, 245–298 (2008) 
[arXiv:0407205 [math.RT]].


[CFM12]Joao1
L. S. Cirio and J. Faria Martins,
“Categorifying the Knizhnik-Zamolodchikov connection,”
Differential Geom. Appl. 30, no. 3, 238–261 (2012)
[arXiv:1106.0042 [hep-th]].


[CFM15]Joao
L. S. Cirio and J. Faria Martins,
“Infinitesimal 2-braidings and differential crossed modules,”
Adv. Math. 277, 426–491 (2015)
[arXiv:1309.4070 [math.CT]].


[CFM17]Joao2
L. S. Cirio and J. Faria Martins,
“Categorifying the 𝔰𝔩(2, ℂ) Knizhnik-Zamolodchikov 
connection via an infinitesimal 2-Yang-Baxter operator in the string Lie-2-algebra,”
Adv. Theor. Math. Phys. 21, 147–229 (2017)
[arXiv:1207.1132 [hep-th]].


[CF94]CraneFrenkel
L. Crane and I. B. Frenkel, 
“Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases,” 
J. Math. Phys. 35, no. 10, 5136–5154 (1994) 
[arXiv:9405183 [hep-th]]. 


[Dri90]Drinfeld90 
V. G. Drinfeld, 
“On quasitriangular quasi-Hopf algebras and on a group that is closely 
connected with Gal(ℚ/ℚ),”
Algebra i Analiz 2, no. 4, 149–181 (1990);
Leningrad Math. J. 2, no. 4, 829–860 (1991).


[ES02]EtingofSchiffmann
P. Etingof and O. Schiffmann, 
Lectures on Quantum Groups, 
Lectures in Mathematical Physics, International Press, Somerville, MA (2002).


[FW20]Fresse
B. Fresse and T. Willwacher,
“The intrinsic formality of 𝔼_n-operads,”
J. Eur. Math. Soc. (JEMS) 22, no. 7, 2047–2133 (2020)
[arXiv:1503.08699 [math.AT]].


[JY21]Yau
N. Johnson and D. Yau,
2-dimensional Categories,
Oxford University Press, Oxford (2021)
[arXiv:2002.06055 [math.CT]].


[Kas95]Kassel
C. Kassel,
Quantum Groups,
Grad. Texts in Math. 155,
Springer-Verlag, New York (1995).


[Kel82]Kelly
G. M. Kelly,
Basic concepts of enriched category theory,
London Math. Soc. Lecture Note Ser. 64,
Cambridge University Press, Cambridge-New York (1982).
[Reprints in Theory and Applications of Categories 10, 1–136 (2005).]


[KL09]KhovanovLauda
M. Khovanov and A. D. Lauda, 
“A diagrammatic approach to categorification of quantum groups. I,” 
Represent. Theory 13, 309–347 (2009) 
[arXiv:0803.4121 [math.QA]].


[KS22]Schnitzer
A. Kraft and J. Schnitzer,
“An introduction to L_∞-algebras and their homotopy theory,”
arXiv:2207.01861 [math.QA].


[Maj95]Majid
S. Majid,
Foundations of Quantum Group Theory,
Cambridge University Press, Cambridge (1995). 


[Maj12]Majid2
S. Majid,
“Strict quantum 2-groups,”
arXiv:1208.6265 [math.QA].


[PTVV13]PTVV
T. Pantev, B. Toën, M. Vaquié and G. Vezzosi,
“Shifted symplectic structures,”
Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)
[arXiv:1111.3209 [math.AG]].


[Pri17]Pridham
J. P. Pridham,
“Shifted Poisson and symplectic structures on derived N-stacks,”
J. Topol. 10, no. 1, 178–210 (2017)
[arXiv:1504.01940 [math.AG]].


[Pri18]PridhamOutline
J. P. Pridham,
“An outline of shifted Poisson structures and deformation quantisation in derived differential geometry,”
arXiv:1804.07622 [math.DG].


[Rie14]Riehl
E. Riehl,
Categorical Homotopy Theory,
New Math. Monogr. 24,
Cambridge University Press, Cambridge (2014).


[RW15]Wagemann
S. Rivière and F. Wagemann,
“On the string Lie algebra,”
Algebr. Represent. Theory 18, no. 4, 1071–1099 (2015)
[arXiv:1407.6806 [math.QA]].


[Rou08]Rouquier
R. Rouquier, 
“2-Kac-Moody algebras,”
arXiv:0812.502 [math.RT].


[Saf21]Safronov
P. Safronov,
“Poisson-Lie structures as shifted Poisson structures,”
Adv. Math. 381, Paper No. 107633 (2021)
[arXiv:1706.02623 [math.AG]].









 Paper Title: Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases.



Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================



Accessibility Statement: a WCAG2.1AA compliant version of this PDF will be posted at https://www.math.tamu.edu/undergraduate/research/REU/the Texas A&M University Mathematics REU Website. We thank the National Science
Foundation’s Office for Strategic Initiatives on ‘STEM Access for Persons with Disabilities’ for funding the creation of the accessible version.



§ ABSTRACT
 
The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective <cit.>. From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of _q(𝔰𝔬_6), using crystal bases to analyze and manipulate various representations of 𝒰_q(𝔰𝔬_6). We conclude that both approaches produce different processes and therefore the previous method of <cit.>, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.



§ INTRODUCTION
 
<cit.> is the first paper considering two-class ASEP. Later research by <cit.> shows that generators of multi-class interacting particle systems can be constructed using a central element from various Drinfeld-Jimbo quantum subgroups of _q(𝔤𝔩_n) <cit.>. In <cit.>, the probabilistic procedure of stochastic fusion is developed and it is shown to produce the same process as central elements on normal ASEP. We focus on an interacting particle system, labeled the Type D ASEP <cit.>, that is a generalization of Spitzer's ASEP <cit.>. The Type D ASEP involves two classes of particles jumping on a lattice, where two particles of different classes can exist at a site but two particles of the same class cannot. The jump rate of an individual particle depends on its class and configuration in relation to other particles.

We question whether constructing a generator according to the probabilistic method of stochastic fusion defined in <cit.> will result in the same generator as the algebraic method, as studied in <cit.>, of applying a ground state transformation on a Casimir element in the second tensor power of an irreducible representation of _q(_6). We find that, unlike in Type A Lie algebras, this is not the case for all Type D Lie algebras; in fact, the probabilistic generator allows for nine states at each lattice site while the algebraic generator allows for fourteen. We therefore explore various characteristics of the different processes generated using these separate approaches. The probabilistic characteristics include limits of component generator matrices, stationary distributions, spectral gaps, and Markov self-duality. In terms of algebraic characteristics, we study the algebraic structure of various representations needed to construct the generator, creating crystal bases and Young tableaux to decompose _q(_6)-modules into irreducible representations as well as weight spaces, then reflecting on the impacts these decompositions have on the eigenvalues and overall structure of the constructed Markov generator.

The paper proceeds by the following outline: algebraic and probabilistic notation is introduced in the remainder of [section–Intro]Section 2, probabilistic and algebraic results are stated in [section–ProbResults]Section 3 and [section–AlgResults]Section 4, respectively, and the respective proofs are discussed in [section–ProbProofs]Section 5 and [section–AlgProofs]Section 6. We list a variety of matrices and methods of calculating them in the [section–Appendix]Appendix.

Acknowledgements. This research was conducted during a mathematics REU at Texas A&M University, funded by the National Science Foundation under REU Site Grant DMS-2150094 and the National Science Foundation’s Office for Strategic Initiatives on ‘STEM Access for Persons with Disabilities’. The authors express their gratitude to their mentor, Dr. Jeffrey Kuan, and teaching assistant, Kennedi Hatcher, for their involvement in the project. 



 §.§ Probability Notation
 


  §.§.§ Type D ASEP


The Type D ASEP is a Markov process in which there are two classes of particles jumping on a lattice. Two particles of the same class cannot share a lattice site, but two particles of different classes can. The Type D ASEP is determined by parameters (q,n,δ = 0). The variable n measures drift speed, with larger values of n indicating more rapid drifts in the Type D ASEP. The variable q represents drift direction, with rightwards drift for 0<q<1 and leftwards drift for q > 1. Note that q cannot equal 1. The variable δ represents the interaction between the two particle classes. The jump rate of a particle in the Type D ASEP is determined by the particle's class and configuration on the lattice.

We use Γ to denote lattice sites of the Type D ASEP. For each Γ site, we use 0 to denote an empty site (contains no particles), 1 to denote a particle of class 1 at a site, 2 to denote a particle of class 2, and 3 to denote a particle of class 1 and a particle of class 2.  Furthermore, we use parentheses to denote a Γ state in the Type D ASEP. For example, we let (3,1,0,2) denote the four Γ state particle configuration in the top line of Figure <ref>.

We now consider the Type D ASEP over two Γ lattice sites with δ = 0. See <cit.> for a rigorous definition and <cit.> for a picture of the generator of Type D ASEP on two Γ lattice sites. Note that there are sixteen (a,b) possible states since 0 ≤ a,b ≤ 3. We use L_p^(2) to denote the generator for Type D ASEP on two Γ sites, which is a direct sum of 

    L_1 = [                               *     (- q^1 - n + q^n - 1)^2/q^2 - q^2 n - 4 + q^2 n - 2 + 2/q^2;           q^2 n - q^2 n - 2 + 2                               *       - q^2 - 2 n + 2 + q^- 2 n         (- q^1 - n + q^n - 1)^2;      q^2(- q^1 - n + q^n - 1)^2   2 q^2 + q^2 - 2 n - q^4 - 2 n                               *   2 q^2 + q^2 - 2 n - q^4 - 2 n;           q^2 n - q^2 n - 2 + 2         (- q^1 - n + q^n - 1)^2       - q^2 - 2 n + 2 + q^- 2 n                               * ]

corresponding to the communicating classes {(3,0),(2,1),(0,3),(1,2)}, four 2 by 2 blocks

    L_2 = [                         *   q^1 - 2 n + q^2 n - 1/q; q (q^1 - 2 n + q^2 n - 1)                         * ]

corresponding to the communicating classes {(1,0),(0,1)},{(2,0),(0,2)},{(3,1),(1,3)},{(3,2),(2,3)}, and four 1 by 1 blocks of zeros corresponding to the communicating classes {(0,0)},{(1,1)},{(2,2)},{(3,3)}. Finally we express L_p^(2),

    L_p^(2) = L_1 ⊕⊕_i=1^4 L_2 ⊕⊕_i=1^4 [0]

with respect to the ordered basis

    Ω = ( (3,0),(2,1),(0,3),(1,2),(1,0),(0,1),(2,0),(0,2),(3,1),(1,3),(3,2),(2,3),(0,0),(1,1),(2,2),(3,3) ).

Futhermore, we expand the definition of Type D ASEP to K lattice sites where the generator matrix is given by

    L = L^1,2+L^2,3+...+L^K-1,K

We let L^x,x+1 denote the matrix the matrix on lattice sites x and x+1. For this paper, we almost always restrict to the case of Type D ASEP on 4 lattice sites. Therefore we define,

    L_p  L^1,2+L^2,3+L^3,4

to represent the generator of Type D ASEP on four Γ sites. The generator L_p is a 256 by 256 matrix, 256 = 4^4. Furthermore, we define L_m to be the generator where only the two middle particles interact, ie. L_m  L^2,3.



  §.§.§ Stochastic Fusion

Taking a lattice with an even number of sites, we define stochastic fusion as the merging of each set of two consecutive lattice sites, such that the particles in each original site of the set are stacked at the new site. Note that two particles of the same species can exist at one new site, and that the total number of sites involved in the process is decreased by one half through stochastic fusion. See <cit.> for a thorough treatment of stochastic fusion.
We proceed to define notation for the fused Type D ASEP, using γ to denote lattice sites and states that have undergone the process of fusion, and Γ to denote lattice sites and states that have not. For γ lattice  sites, we use 11 to denote two particles of class 1 at a site, 22 to denote two particles of class 2, 3 to denote a particle of class 1 and a particle of class 2, 33 to denote two particles of class 1 and two particles of class 2, 31 to denote two particles of class 1 and one particle of class 2, and 32 to denote one particle of class 1 and two particles of class 2.  We will use angle brackets to denote γ states. See Figure <ref> for a visualization of this notation. As another example of γ-state notation, note that ⟨ 31,2 ⟩ is the γ state of (3,1,0,2) and is depicted in the bottom line of Figure <ref>.





Let S be the state space for the four-site Type D ASEP, in which there are 256 = 4^4 possible states. Rigorously, 
    S = {(a, b, c, d) | a, b, c, d ∈{0,1,2,3}}
 Fusing our lattice, we reach the state space Ŝ of two γ lattice  sites, in which there are 81 = 9^2 possible states. Rigorously, 
    Ŝ = {⟨ x, y ⟩| x, y ∈{0,1,2,3,11,22,31,32,33}}


Our generator, fusion, and fission matrices depend on the ordering of states, so we order states in the Type D ASEP over two Γ lattice  sites as Ω and we order states in the Type D ASEP over one γ lattice site as 


    χ = (  0 , 1 ,  2 ,  11 ,  12 ,  22 , 31 , 32 ,  33 )
 
 A visualization of this ordering and the possible states over one γ lattice site can be seen in Figure <ref>. See section <ref> for the ordering of Γ and γ states over four Γ lattice sites and two γ lattice sites, respectively.

We now formally define the stochastic fusion map.



We define ϕ:Ω→χ as the stochastic fusion map for the Type D ASEP on two Γ lattice  sites, where x_i ∈{0,1,2,3} describes the particles at Γ lattice site i:

    ϕ(x_1,x_2)  0     x_1,x_2 = 0 
     x_1     x_1 ≠ 0, x_2 = 0 
     x_2     x_1 = 0, x_2 ≠ 0  
     x_1x_2     x_1,x_2 ≠ 0, x_1 ≥ x_2 
     x_2x_1     x_1,x_2 ≠ 0, x_1 ≤ x_2



This definition is implied as a specific case in <cit.> when there are only class 1 and class 2 particles. Notice that this map corresponds to stacking particles where order does not matter. To perform stochastic fusion on four Γ lattice sites of the Type D ASEP the stochastic fusion map is ϕ⊗ϕ, and so on for additional sites of stochastic fusion.



  §.§.§ Transition and Generator Matrices


We now turn to the transition matrices of the Type D ASEP. Let P_t be the transition matrix and L_p be the generator matrix for the Type D ASEP on four Γ sites defined previously. Note that L_m, the generator matrix for the Type D ASEP on four Γ lattice sites where only the two middle Γ lattice sites interact, is defined in <ref>. Fusing down to two γ lattice sites, Q_t is the fused transition matrix and L_Q is the generator matrix for the Type D ASEP on two γ lattice  sites. Summarizing this process, we let Λ: Ŝ→ S be the matrix of the fission map from the Type D ASEP on two γ lattice  sites to Type D ASEP on four Γ sites, and Φ: S →Ŝ be the matrix of the stochastic fusion map from the Type D ASEP on four Γ lattice sites to two γ sites. See Figure <ref> for an example of Λ and Φ mappings.

We also define notation for the two-Γ site Type D ASEP. Let Λ^(2) be the matrix of the fission map from the Type D ASEP on two Γ lattice  sites to one γ site, and Φ^(2) be the corresponding matrix of the stochastic fusion map.




 §.§ Algebraic Notation

We must first define _q(_6) as well as its fundamental representation in order to construct more complex _q(_6)-modules. We introduce crystal base theory as needed in Section <ref>.

    The special orthogonal Lie group SO_6 is the multiplicative group with elements of the form
    
    SO_6 = { X ∈ M_6 × 6 (): XX^T = I,  X = 1}




    The special orthogonal Lie algebra _6 is the Lie algebra with elements of the form
    
    𝔰𝔬_6 = {[    A    C; -C^T    B;      ]: A,B,C ∈ M_3 × 3 (),  A = - A^T, B=-B^T}





    The Universal Enveloping Algebra of _6, denoted (_6), is generated by 

    {E_1, E_2, E_3, F_1, F_2, F_3, H_1, H_2, H_3} and the following relations:
    
    [E_i, F_i] = H_i,  1 ≤ i ≤ 3

    and
    
    E_l^2E_j + E_jE_l^2 = 2E_lE_jE_l;  F_l^2F_j + F_jF_l^2 = 2F_lF_jF_l

    for (l,j) ∈{ (1,2), (2,1), (1,3), (3,1) }.



    The q-deformed quantum group _q(_6), as described by <cit.> and <cit.>, is generated by {E_1, E_2, E_3, F_1, F_2, F_3, q^H_1, q^H_2, q^H_3} and the relations:
    
    [E_i, F_i]    = q^H_i - q^-H_i/q-q^-1
    
        q^H_iE_j    = q^α_i ·α_jE_j q^H_i
    
        q^H_iF_j    = q^-α_i ·α_jF_j q^H_i

    for 1 ≤ i,j ≤ 3 and 
    
    E_l^2E_k + E_kE_l^2 = (q+q^-1)E_lE_kE_l;  F_l^2F_k + F_kF_l^2 = (q+q^-1)F_lF_kF_l

    for (l,k) ∈{ (1,2), (2,1), (1,3), (3,1) }.
    
    For a matrix H in the Cartan subalgebra 𝔥 of _6, let L_i ∈𝔥^* map H to H_i,i. Then we define the simple roots of _q(_6) to be
        
    Π = {α_1 = L_1 - L_2, α_2 = L_2 - L_2, α_3 = L_2 + L_3}

    and the fundamental weights to be
        
    {ω_1 = L_1, ω_2 = 1/2(L_1+L_2-L_3), ω_3 = 1/2(L_1+L_2+L_3) }.



 
    We define the coproduct Δ, counit ϵ, and antipode S of _q(_6) to be
    
    Δ (E_i) = E_i ⊗ 1 + q^H_i⊗ E_i, ϵ(   E_i) = 0, S(E_i) = -E_iq^H_i
    Δ(F_i) = 1 ⊗ F_i + F_i ⊗ q^-H_i, ϵ(   F_i) = 0, S(F_i) = -q^-H_iF_i 
    Δ(q^H_i) = q^H_i⊗ q^H_i,ϵ(   q^H_i) = 1,
            S(q^H_i) = q^-H_i.

    Equipping _q(_6) with Δ, ϵ, and S makes the quantum group a Hopf algebra.




 
    A fundamental representation of 𝒰_q(𝔰𝔬_6) is defined to be the subset of M_6 × 6([q,q^-1]) as follows: 

    
    
    E_i,j indicates the 6× 6 zero matrix with a 1 in the (i,j)^th entry, and q^-H_i is defined as the multiplicative inverse of q^H_i. We denote V to be ^6 ⊗^6.





§ PROBABILISTIC RESULTS



We begin by defining the generator for stochastic fusion on the Type D ASEP, noting the block diagonal form of this matrix. We then examine the unique characteristics of the generator's component blocks (communicating classes), particularly their limits and spectral gaps as drift speeds are taken to infinity and their stationary distributions in relation to drift speed. Fixing drift speed to two, we explore two approaches to finding the matrix of Markov self-duality for our system. The matrix of Markov self-duality will be expressed in terms of the eigenvectors of the block-diagonalized fusion generator, allowing for the dual and spectral gap to be found through the same eigenvalue-eigenvector calculation. 



 §.§ Producing Stochastic Fusion Matrix

In this section we perform stochastic fusion on Type D ASEP for four Γ lattice sites following the procedure developed in Section 3 of <cit.>. To begin, we define 

    Q_t Λ P_t Φ

Taking the derivative on both sides at zero, we express the same equation in terms of generators.

    L_Q = Λ L_p Φ

The goal of this section is to explicitly calculate L_Q, the generator matrix for two γ sites. This process reduces the 256 × 256 matrix L_p to the fused 81 × 81 matrix L_Q.

We construct Λ^(2) based on the reversibility measures given in Proposition 1.3 of <cit.> and we define ΛΛ^(2)⊗Λ^(2) in <ref>. Similarly, we construct Φ^(2) based on the map ϕ from two Γ lattice sites to a single γ lattice site, and we define ΦΦ^(2)⊗Φ^(2) in <ref>.

The generator matrix L_m corresponds to only the two middle of the four Γ lattice sites interacting, and can be expressed as a permutation of Id_16 × 16⊗ L_p^(2) as shown in <ref>, this stands in contrast to L_p where all four Γ sites can interact. We show that only considering when the two middle of the four Γ sites interact will lead to the same stochastically fused process L_Q.

[]theoremMiddleSwap

The generator for Type D ASEP on two γ sites denoted L_Q is equal to Λ L_m Φ, where L_m is the generator for Type D ASEP on four Γ sites where only the middle two Γ sites interact.


See a proof of this theorem in section <ref>.

Next, we want to understand the communicating classes of L_Q. We show that L_Q can be broken up into twenty-five communicating classes, where several of the generators on these twenty-five communicating classes are the same and every communicating class with the same amount of states has the same generator. We define L^Q_D as a permutation of L_Q that is a block diagonal matrix, where the ordering is based on the communicating classes.

[]propositionDiagonalBlockOrdering

Let L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0], where ℒ_9 is the generator for the 9-state communicating class, ℒ_6 is the generator for each 6-state communicating class, and so on for ℒ_4,ℒ_3 and ℒ_2. There exists a permutation matrix C such that L_Q = CL^D_QC^-1.

The permutation of states from L_Q to the block diagonal L^D_Q which defines the permutation matrix C is listed in <ref>. Below are the four three-state communicating classes of L_Q: 
    { 1,1 ,  0,11 ,  11,0 }, { 2,2 ,  0,22 ,  22,0 }, { 31,31 ,  11,33 ,  33,11 }, { 32,32 ,  22,33 ,  33,22 }
.

    ℒ_3 =  [                           * q^2+q^4n/(q^6+2q^4+q^2)q^2n q^6+q^4n+4/(q^4+2q^2+1)q^2n;               q^2+q^4n/q^2n                           *                           0;             q^2+q^4n/q^2n+2                           0                           * ]

The remaining ℒ_9,ℒ_6,ℒ_4,ℒ_2 matrices are outlined in section <ref>.



 §.§ Taking n to Infinity and Spectral Gaps


Recall that n affects drift speed, with larger values of n causing faster drifting in the Type D ASEP. Also recall that Type D ASEP particles drift to the right for 0<q<1 and to the left for q > 1. We are interested in how the drift speed of the Type D ASEP affects the drift speed of the stochastically fused process and its relaxation times.

Faster drift speeds in the Type D ASEP correspond to faster drift speeds in the fused Type D ASEP process. Recall that L_m is the generator for the Type D ASEP on four lattice sites where only the two middle Γ sites interact, and note that L_m depends on n. Thus, increasing values of n will indicate faster drift speeds for the Type D ASEP on four Γ lattice sites where only the two middle Γ sites interact. We see that Λ and Φ do not depend on n, so faster drift speeds in L_m will lead to faster drift speeds in L_Q, the fusion matrix. This statement is proved rigorously below.

[]lemmacommutativityProposition

We have the following limits: lim_n →∞ L_Q = lim_n →∞ (Λ L_m Φ) = Λ (lim_n →∞ L_m) Φ. In other words, the limit as n →∞ is commutative on L_Q.


[]propositionfiniteProposition

The following are finite: lim_n →∞ (q^-2n L_m) for q >1 and lim_n →∞ (q^2n L_m) for 0 < q < 1.



[]theoremfiniteLimitTheorem

The following are finite: lim_n →∞ (q^-2n L_Q) for q > 1, and lim_n →∞ (q^2n L_Q) for 0 < q < 1.


See a proof of this theorem in section <ref>.

The previous theorem proves the intuitive statement that, as the drift speed of the Type D ASEP on four Γ lattice sites increases, so does the drift speed of the stochastically fused process.

Continuing our study of limits as n approaches infinity, we now consider the limits of spectral gaps. The spectral gap of a Markov process is the absolute value of the second-largest (ie. least negative) eigenvalue of the generator matrix. Note that eigenvalues of generator matrices are less than or equal to zero. From our calculations, the spectral gaps corresponding to generators ℒ_2 and ℒ_3 simplify to reasonable expressions, while the spectral gaps for ℒ_4,ℒ_6, and ℒ_9 do not simplify to expressions of a reasonable length. 


    
    The spectral gaps for ℒ_2 and ℒ_3 communicating classes are denoted in the form |λ_ℒ_i| and are:
    
    |λ_ℒ_2| = (q^4n-2+1)(q^4+1)/q^2n(q^2+1)


    |λ_ℒ_3|   =  | -(q^4n-2+1)(2q^4+q^2+2)/2q^2n(q^2+1) +(q^2 + q^4n)/q^2n(q^2+1) | = (q^4+1)(q^2+q^4n)/q^2n+2(q^2+1).



For q > 1 and 0 < q < 1, notice that the spectral gaps |λ_ℒ_2| and |λ_ℒ_3| go to infinity as n goes to infinity, meaning the relaxation times 1/λ_ℒ_2 and 1/λ_ℒ_3 go to zero. This implies that as the drift speed of the Type D ASEP goes to infinity, the time to convergence to the stationary distribution approaches zero for the 2 and 3-state communicating classes. The authors suspect the same is true for the 4, 6, and 9-state communicating classes.

Note that, for i ∈{ 2, 3 }, the eigenvalues of q^± 2nℒ_i are those of ℒ_i multiplied by q^± 2n. Thus, the spectral gaps of  q^± 2nℒ_2 and q^± 2nℒ_2 are |q^± 2nλ_ℒ_2| and |q^± 2nλ_ℒ_3|, respectively. Note that the following spectral gaps and relaxation times are the same for the 2 and 3-state communicating classes. The limit as n goes to infinity of the spectral gap of q^-2nℒ_i for q > 1 is 1. The limit as n goes to infinity of the spectral gap of q^2nℒ_i for 0 < q < 1 is q^4 + 1/q^2 + 1. Thus, the relaxation times for both generators for q > 1 are 1, and the relaxation times for 0 < q < 1 are q^2 + 1/q^4 + 1. This result suggests that the time to convergence to the stationary distribution increases (relative to the time to convergence in the previous paragraph) as n goes to infinity for the 2 and 3-state communicating classes. The authors suspect the same is true for the 4, 6, and 9-state communicating classes.


As n goes to infinity, some of the jump rates in the (time-rescaled) fused Type D ASEP converge to 0, which can be seen by taking the limit as n →∞ of entries in the fusion generator. Thus, as n approaches infinity, the fused Type D ASEP (where δ = 0 and time is rescaled by a factor of q^± 2n for respective values of q) degenerates to the usual ASEP. In this degeneration, we would expect the 2-state communicating class and the 3-state communicating class to have the same relaxation time, which is consistent with the calculations in the previous paragraph.     


Due to the difficulty of displaying the eigenvalues for general q and n, we note that for n=2, the eigenvalues simplify nicely and are used to find a dual for L_Q when n=2 in <ref>.

[]lemmaLeigenvalues

The eigenvalues for ℒ_9, ℒ_6, ℒ_4, ℒ_3, and ℒ_2 when n = 2 are the eigenvalues of L_Q^D for n = 2.





 §.§ Stationary Distributions
 

Take Markov process X_t with generator matrix L and stationary distribution π. Then π L = 0. In other words, π is a left eigenvector of L. 
We use this result to obtain Proposition <ref>, which depends on the communicating classes listed in Section <ref>.

[]propositionallStationary 
For the 9-state communicating class, the stationary distribution is the normalization of 

    [q^4(q^2+1)^4,1,q^16,q^2(q^2+1)^2,q^10(q^2+1)^2,q^2(q^2+1)^2,q^10(q^2+1)^2,q^8,q^8].


For the 6-state communicating class, the stationary distribution is the normalization of 

    [q^2(q^2+1)^2,q^6(q^2+1)^2,1,q^12,q^4,q^8].


For the 4-state communicating class, the stationary distribution is the normalization of 

    [q^8,q^4,q^4,1].


For the 3-state communicating class, the stationary distribution is the normalization of 

    [q^2(q^2+1)^2,1,q^8].


For the 2-state communicating class, the stationary distribution is the normalization of 

    [1, q^4].


for all q > 0 where q ≠ 1. 



Our advisor, Dr. Jeffrey Kuan, noticed that the stationary distributions we found resemble q-exchangeable measures <cit.>, but we do not delve into the details of this relationship in our paper.


In order to find the stationary distributions, we found the left eigenvectors corresponding to zero for n = 2 (drift speed 2), and then verified that the same eigenvectors work for general n. We then normalized the vectors to obtain our final stationary distributions. This emphasizes that, although the jump rates of particles in the Type D ASEP do depend on n, the stationary distributions do not.



 §.§ Markov Duality



Let X_t be a time-homogeneous Markov process on a discrete state space 𝒳. Label the generator matrix of the Markov process as L_X. Let D be a matrix with rows and columns indexed by 𝒳. Then X_t is self-dual with respect to the matrix D if L_X D = D L_X^T
<cit.>.




  §.§.§ Duality with Quantum q-Krawtchouk Polynomials
 

Our mentor, Dr. Jeffrey Kuan, previously dropped the term “quantum” from “quantum q-Krawtchouk polynomials” <cit.>. We have used the correct term in our paper. 


There are several papers on quantum q-Krawtchouk polynomials, see <cit.> for example. Furthermore, in Theorem 3.1 of <cit.>, the authors propose a self-duality function for the unfused Type D ASEP with parameters (q, n = 2, δ = 0) and (q, n = 3, δ = 0). We question whether their self-duality function works for L_Q^D, the generator of the Type D ASEP over two γ lattice sites with ordered state space 𝒳_2 and parameters (q, n = 2, δ = 0), when approached from a probability perspective.

Using the notation of <cit.> with minimal changes for clarity, we denote the self-duality function given by <cit.> as D_α_1, α_2^(L) (η, ξ) = D_α_1^(L) (η_1, ξ_1) · D_α_2^(L) (η_2, ξ_2) where D_α_i^(L) (η_i, ξ_i) is taken over particle class i ∈{1,2}. We take variables α_1, α_2 ∈ (0, q^4). Using the parameters L = 2 and  n = 2, we find a potential self-duality function (given by Theorem 3.1 in <cit.>) for this 2-site Type D ASEP, where D is the matrix of this proposed self-duality function. For each 𝒳_2[i] = η and 𝒳_2[j] = ξ, the (i,j)th entry of D is given by  D_α_1, α_2^(2) (η, ξ).

However, using the matrix D that we computed, L_Q^D D[1,2] ≠ D(L_Q^D)^T[1,2], so L_Q^D D ≠ D(L_Q^D)^T and thus D is not the matrix of the self-duality function of the Type D ASEP with two γ lattice sites. In our case, 
constructing our particle system from a probabilistic rather than an algebraic process <cit.>, resulted in our particle systems having different duality functions.



  §.§.§ Other Avenues to Duality


Since the aforementioned duality function does not generate a valid dual for the stochastically fused Type D ASEP, another logical approach is to find an algebraic connection between the dual of the Type D ASEP generator on two Γ lattice sites and the dual of the stochastically fused process on two γ lattice sites. Unfortunately, we were not able to find a direct algebraic relation between a dual of L_p^(2) and a dual of L_Q.

Due to the eigenvalues of L_Q not simplifying to reasonable expressions, we now restrict to the case where n = 2 and find a dual based on the eigenvectors. We construct a non-trivial dual for L^D_Q, a permutation of L_Q in block diagonal form defined in <ref>, for n = 2. Note that the eigenvalues for the communicating classes of L_Q^D when n=2 were already given in <ref>.

Recall the structure of the block diagonal generator in <ref>:

    L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0].

Since each communicating class is diagonalizable, we diagonalize L^D_Q block by block and express L_Q^D in a diagonalized form.

[]lemmaLdiagonalizable


    
    
  a. The block generators ℒ_9, ℒ_6, ℒ_4, ℒ_3, ℒ_2 are diagonalizable for n = 2. Let 𝒫_i denote the matrix of the right eigenvectors for the diagonalization of ℒ_i, and let 𝒜_i be the diagonal matrix of eigenvalues listed in <ref> such that ℒ_i = 𝒫_i 𝒜_i 𝒫^-1_i for i ∈{2,3,4,6,9}.
    
  b. Define 𝒫 = 𝒫_9 ⊕⊕_i=1^4 𝒫_6 ⊕⊕_i=1^4 𝒫_4 ⊕⊕_i=1^4 𝒫_3 ⊕⊕_i=1^8 𝒫_2, 𝒜 = 𝒜_9 ⊕⊕_i=1^4 𝒜_6 ⊕⊕_i=1^4 𝒜_4 ⊕⊕_i=1^4 𝒜_3 ⊕⊕_i=1^8 𝒜_2 and 𝒵 = ⊕_i=1^4 [0].
    
  c. The block diagonal matrix L_Q^D = 𝒫𝒜𝒫^-1⊕𝒵.




[]theoremDiagonalStrategy

The matrix 𝒟 = 𝒫𝒫^T ⊕𝒵 is a non-trivial dual for L_Q^D.

See a proof of this theorem in <ref>.

Since 𝒫 has a block diagonal form, it follows that 𝒫𝒫^T has a block diagonal form,

    𝒟 = 𝒫(𝒫)^T ⊕𝒵 = 𝒫_9(𝒫_9)^T ⊕⊕_i=1^4 𝒫_6(𝒫_6)^T ⊕⊕_i=1^4 𝒫_4(𝒫_4)^T ⊕⊕_i=1^4 𝒫_3(𝒫_3)^T ⊕⊕_i=1^8 𝒫_2(𝒫_2)^T ⊕𝒵.

Note that 𝒟 is a non-trivial dual because 𝒟 is not a constant matrix. For example

    𝒫_4(𝒫_4)^T =
    [          1 + 1/q^8 + 1/q^16    1 - q^4 - 1/q^8 - 1/q^12                  1 - 1/q^12           1 - 1/q^4 + 1/q^8;    1 - q^4 - 1/q^8 - 1/q^12 2 + (q^4 - 1)^2/q^8 + 1/q^8                       1/q^8     1 + q^4 - 1/q^4 - 1/q^4;                  1 - 1/q^12                       1/q^8                   2 + 1/q^8                   1 - 1/q^4;           1 - 1/q^4 + 1/q^8     1 + q^4 - 1/q^4 - 1/q^4                   1 - 1/q^4                           3 ].







§ ALGEBRAIC RESULTS
 
Previous research used the symmetries of Type A, C, and D quantum groups to construct asymmetric interacting particle systems on various lattice sites <cit.>. The main algebraic focus of this project was to investigate how this construction differs for the Type D Lie algebra _6. We are also interested in whether this method produced the same Markov generator as the probabilistic approach did. We found the following key result, with the second research question answered in Proposition <ref>. Our definition of a ground state transformation can be found in Definition <ref>.
 
    A Markov generator is produced by performing a ground state transformation on the representation W ⊗ W of <cit.>'s Casimir element; the explicit generator can be found in Section <ref>.

To prove this, we need to first construct the aforementioned representation of _q(_6). We therefore prove the following two propositions.
[]propositionpropositionWtwentydim

    Define W to be the 20-dimensional subspace of ^6 ⊗^6 satisfying 
    Sym_q^2(^6) = W ⊕span{q^-2e_1 ⊗ e_4 + q^2 e_4 ⊗ e_1 +q^-1e_2⊗ e_5 + qe_5⊗ e_2+ e_3 ⊗ e_6 + e_6 ⊗ e_3 }
  with {e_i| i = 1,...,6} standard basis vectors of ^6. Then, W is an irreducible representation of _q(_6).


    As a representation of _q(_6), W ⊗ W decomposes into a direct sum of irreducible representations and thus a direct sum of weight spaces, listed respectively as follows:
         
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0)

         Let ⟨ i,j,k ⟩ denote the weight space W ⊗ W [i,j,k]. Then
         
    W ⊗ W ≅   ⟨0,0,0⟩⊕⟨1,1,0⟩⊕⟨1,0,1⟩⊕⟨0,1,1⟩⊕⟨-1,1,0⟩⊕⟨-1,0,1⟩⊕⟨0,-1,1⟩⊕⟨1,-1,0⟩
       ⊕⟨1,0,-1⟩⊕⟨0,1,-1⟩⊕⟨-1,-1,0⟩⊕⟨-1,0,-1⟩⊕⟨0,-1,-1⟩⊕⟨2,0,0⟩⊕⟨0,2,0⟩
       ⊕⟨0,0,2⟩⊕⟨-2,0,0⟩⊕⟨0,-2,0⟩⊕⟨0,0,-2⟩⊕⟨2,1,1⟩⊕⟨1,2,1⟩⊕⟨1,1,2⟩⊕⟨2,1,-1⟩
       ⊕⟨2,-1,1⟩⊕⟨1,2,-1⟩⊕⟨-1,2,1⟩⊕⟨1,-1,2⟩⊕⟨-1,1,2⟩⊕⟨2,-1,-1⟩⊕⟨-1,2,-1⟩
       ⊕⟨-1,-1,2⟩⊕⟨-2,1,1⟩⊕⟨1,-2,1⟩⊕⟨1,-2,1⟩⊕⟨-2,1,-1⟩⊕⟨-2,-1,1⟩⊕⟨1,-2,-1⟩
       ⊕⟨-1,-2,1⟩⊕⟨1,-1,-2⟩⊕⟨-1,1,-2⟩⊕⟨-2,-1,-1⟩⊕⟨-1,-2,-1⟩⊕⟨-1,-1,-2⟩
       ⊕⟨2,2,0⟩⊕⟨2,0,2⟩⊕⟨0,2,2⟩⊕⟨-2,2,0⟩⊕⟨-2,0,2⟩⊕⟨0,-2,2⟩⊕⟨2,-2,0⟩⊕⟨2,0,-2⟩
       ⊕⟨0,2,-2⟩⊕⟨-2,-2,0⟩⊕⟨-2,0,-2⟩⊕⟨0,-2,-2⟩⊕⟨3,1,0⟩⊕⟨3,0,1⟩⊕⟨0,3,1⟩
       ⊕⟨1,3,0⟩⊕⟨1,0,3⟩⊕⟨0,1,3⟩⊕⟨3,-1,0⟩⊕⟨3,0,-1⟩⊕⟨0,3,-1⟩⊕⟨-1,3,0⟩⊕⟨-1,0,3⟩
       ⊕⟨0,-1,3⟩⊕⟨-3,1,0⟩⊕⟨-3,0,1⟩⊕⟨0,-3,1⟩⊕⟨1,-3,0⟩⊕⟨1,0,-3⟩⊕⟨0,1,-3⟩
       ⊕⟨-3,-1,0⟩⊕⟨-3,0,-1⟩⊕⟨0,-3,-1⟩⊕⟨-1,-3,0⟩⊕⟨-1,0,-3⟩⊕⟨0,-1,-3⟩
       ⊕⟨4,0,0⟩⊕⟨0,4,0⟩⊕⟨0,0,4⟩⊕⟨-4,0,0⟩⊕⟨0,-4,0⟩⊕⟨0,0,-4⟩.


Once we characterized W ⊗ W, we were able to extract enough information to construct a 400 × 400 representation of <cit.>'s Casimir element without significant computation, which brings us to our next proposition. 
 
     We can write π_W⊗ W(C) as a block matrix with one 22× 22 block, twelve 12× 12 blocks, six 8 × 8 blocks, twenty-four 4 × 4 blocks, twelve 3 × 3 blocks, twenty-four 2 × 2 blocks, and six 1 × 1 blocks. Denoting v_i to be the i^th basis vector of W, the blocked matrix is with respect to the ordered basis in Section <ref>.

Finally, we must perform a ground state transformation so that each row of π_W⊗ W(C) sums to 0. We introduce and tweak the method in Section <ref> to compute this transformation. 

[]propositionpropositiongroundstate

    Applying variations of
    
    ⟨ u_i ⊗ u_j| Δ(F_1)^k_1Δ(F_2)^k_2Δ(F_3)^k_3| e_1 ⊗ e_1 ⟩

    will generate zero values for the lowest weight vector of the _q(_6)-modules V, W, and W ⊗ W.

Using the previous four propositions, we compute a 196 × 196 fused Type D ASEP system generator from <cit.>'s central element represented in W ⊗ W, which can be found in Section <ref>. This proves Theorem <ref>, and also has probabilistic significance: as an interacting particle system, the process allows for 14 different states at each site, i.e., all states except four particles of the same class. This differs from the probabilistically-constructed generator, which does not allow for three particles of the same class on a lattice site.

Finally, we compare our result to the fused generator matrix L_Q defined in Section <ref> and written explicitly in Section <ref>.


    Regardless of which ground state transformation is applied to π_W ⊗ W(C), the resulting Markov generator will not match the probabilistically-generated matrix L_Q in Section <ref>. 





§ PROBABILISTIC PROOFS
 


 §.§ Producing Stochastic Fusion Matrix


To begin, consider the function ϕ. The left side of Figure <ref> depicts ϕ(3,1) =  31 and the right side depicts ϕ(0,2) =  2. Note that the function ϕ is surjective.

In order to construct Λ^(2), we recall from <cit.>, that Λ^(2) is dependent on the reversible measures of the process generated by L_p^(2), in other words Type D ASEP on two Γ sites. We define the function G^2 below which is expressed fully in Proposition 1.3 of <cit.> and gives the reversible measures for the Type D ASEP on two Γ lattice sites.

Any particle configuration on two lattice sites can be represented as a function η:{1,2}→{0,1,2,3}, a function from a lattice site to the particle configuration on that lattice site. Furthermore, A_1(η) ⊂{1,2} is the set of lattice sites that class 1 particles occupy and similarly for A_2(η). Finally, below is the function from Proposition 1.3 of <cit.>,

    G^2(η) = ∏_x ∈ A_1(η) q^-2x∏_x ∈ A_2(η) q^-2x

For simplicity, we represent the function η as an ordered pair (η_1,η_2), where η(1) = η_1 and η(2) = η_2. As an example for the reader to check their understanding with the new notation,

    G^2(2,1) = ∏_x ∈{2}q^-2x∏_x ∈{1}q^-2x = q^-6



    
    We construct and express Λ^(2), and define the fission map ΛΛ^(2)⊗Λ^(2).



Let the rows of Λ^(2) be indexed by x ∈χ and the columns be indexed ω∈Ω. We define

    (Λ^(2))_x, ω = 
           
        G^2(ω)    ϕ(ω) = x 
    
        0    ϕ(ω) ≠ x

with the additional step that the rows of Λ^(2) are normalized to sum to 1. Λ^(2) is below.

    [             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1;             0             0             0             0     q^2/q^2+1       1/q^2+1             0             0             0             0             0             0             0             0             0             0;             0             0             0             0             0             0     q^2/q^2+1       1/q^2+1             0             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             1             0             0; q^4/(q^2+1)^2 q^2/(q^2+1)^2   1/(q^2+1)^2 q^2/(q^2+1)^2             0             0             0             0             0             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1             0;             0             0             0             0             0             0             0             0     q^2/q^2+1             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0     q^2/q^2+1             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1 ]




    
    We construct and express Φ^(2), and define the fusion map ΦΦ^(2)⊗Φ^(2).



    
    Let the rows of Φ^(2) be indexed by ω∈Ω and the columns be indexed by x ∈χ. We define
    
    (Φ^(2))_ω , x = 
        
            1    ϕ(ω) = x 
    
            0    ϕ(ω) ≠ x

    Note that Φ^(2) is a deterministic map and Φ^(2) is the matrix representation of ϕ. Φ^(2) is below.
    

    [ 0 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0; 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 0; 0 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 1 0; 1 0 0 0 0 0 0 0 0; 0 0 0 1 0 0 0 0 0; 0 0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0 1;   ]



We now prove that L_m, the generator matrix for the Type D ASEP on four γ lattice sites where only the middle two Γ states interact, can be permuted into _16 × 16⊗ L^(2)_p. 


    
    There exists a permutation matrix J such that L_m = J(_16 × 16⊗ L_p^(2))J^-1. In other words, the generator matrix L_m can be reordered into _16 × 16⊗ L_p^(2).



    
    For a short proof by existence, consider the diagonal ordering of states of four Γ lattice sites listed in <ref>, we permute L_m according to these states and get _16 × 16⊗ L_p^(2).
    
    For a more intuitive proof, consider that any state in the Type D ASEP on four Γ lattice sites can be represented as (x_1,x_2,x_3,x_4) where x_1,x_2,x_3,x_4 ∈{0,1,2,3}. Letting only the middle Γ sites interact corresponds to x_1 and x_4 being held constant, while the interaction of x_2 and x_3 is governed by L_p^(2). Now consider the bijective function h:(x_1,x_2,x_3,x_4) ↦((x_1,x_4),(x_2,x_3)). Since (x_1,x_4) is held constant while (x_2,x_3) interact, it follows that L_m can be permuted into ⊕_i=1^16 L_p^(2), since there are 16 possibilities for (x_1,x_4). Finally, it follows that ⊕_i=1^16 L_p^(2) = _16 × 16⊗ L_p^(2). This proves that L_m can be permuted into _16 × 16⊗ L_p^(2).

Finally we prove <ref>, and therefore that the fused process only depends on the middle interacting particles.
*

    
    We can express L_p = L^1,2+L^2,3+L^3,4. Note that L_m  L^2,3, as stated in <ref>. Now we show that,
    
    L_Q Λ L_p Φ = Λ (L^1,2+L_m+L^3,4) Φ

    
    = Λ L^1,2Φ+Λ L_m Φ+Λ L^3,4Φ

    
    = (Λ^(2)⊗Λ^(2))L^1,2(Φ^(2)⊗Φ^(2)) + Λ L_m Φ + (Λ^(2)⊗Λ^(2))L^3,4(Φ^(2)⊗Φ^(2))

    
    = 0+Λ L_m Φ+0

    
    = Λ L_m Φ

    The second to last step follows because L^1,2 and L^3,4 are the generators acting on the sites that get fused down by Φ and fissioned by Λ. The fused state of the Type D ASEP on two Γ lattice sites which are being fused cannot change because the stochastically fused state is completely dependent on the number of type 1 and type 2 particles on those two Γ lattice sites. This cannot change when only those two sites interact.
    

*

    
    Permute L_Q according to the diagonal ordering for two γ given in <ref>. This is a permutation of L_Q according to its communicating classes.





 §.§ Taking n to Infinity
 

*
 
We remind the reader that Λ and Φ do not depend on n. Therefore,

    lim_n →∞ L_Q = lim_n →∞ (Λ L_m Φ) = Λ (lim_n →∞ L_m) Φ



*


Take q >1. Observing L^(2)_p, we see that multiplying it by q^-2n then taking the limit as n →∞ is finite. The same applies to multiplying L^(2)_p by q^2n for 0 < q< 1.  Finally, use <ref> and notice that

    q^± 2n L_m = q^± 2nJ(_16 × 16⊗ L^(2)_p)J^-1 = J(_16 × 16⊗ (q^± 2n L^(2)_p))J^-1
 for respective boundaries on q, by the properties of the Kronecker product and standard matrix multiplication. Noting that J and J^-1 do not depend on n, we take the limit as n →∞ of both sides and see that lim_n →∞ (q^± 2n L_m) = lim_n →∞ (J(_16 × 16⊗ (q^± 2n L^(2)_p))J^-1) = J(_16 × 16⊗ (lim_n →∞ q^± 2n L^(2)_p))J^-1, where the right (and therefore left) side of the equation is finite since lim_n →∞ (q^± 2n L^(2)_p) is finite for respective boundaries on q. This implies that the rate matrix grows on the order of q^2n irrespective of drift direction.


*


Note that Proposition <ref> suggests that multiplying q^± 2n L_m (for respective boundaries on q) by any constant preserves the finite limit, since the constant can be moved outside the limit. Note that 
    q^± 2n L_Q = q^± 2nΛ L_m Φ = Λ q^± 2n L_m Φ
 by the commutativity of scalar multiplication. Taking the limit of both sides of the equation, we have 
    lim_n →∞ (q^± 2n L_Q) = lim_n →∞ (Λ q^± 2n L_m Φ) = Λlim_n →∞ (q^± 2n L_m) Φ
 by Lemma <ref>. We know that lim_n →∞ (q^± 2n L_m) is finite for respective values of q, so lim_n →∞ q^± 2n L_Q must also be finite for such values of q.


*

We list the eigenvalues for the generator ℒ_9 of the 9-state communicating class:


    0, -q^8 - q^4 - 1/q^4, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4, -q^12 - 4q^10 - 2q^8 - 6q^6 - 2q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4,


    -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_6 of the 6-state communicating class:

    0, -q^8 - q^4 - 1/q^4,  -q^8 + q^6 - 2q^4 + q^2 - 1/q^4,


    -q^12 - 4q^10 - 2q^8 - 6q^6 - 2q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4,  -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_4 of the 4-state communicating class:

    0, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4,  -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_3 of the 3-state communicating class:

    0, -q^8 - q^4 - 1/q^4,  -q^8 + q^6 - 2q^4 + q^2 - 1/q^4

The eigenvalues for the fusion generator ℒ_2 of the 2-state communicating class:

    0, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4





 §.§ Stationary Distribution

*
 
For communicating class i with generator matrix ℒ_i, the stationary distribution for the class is represented by the normalized left eigenvector π corresponding to the eigenvalue zero for the generator matrix, ie. π L_i = 0. Through standard computation, one can see that a left eigenvector exists for each communicating class and is equal to the one listed in the proposition, making the normalization of that vector the stationary distribution for the corresponding communicating class.




 §.§ Markov Duality



*

To prove part a, we show that for each communicating class ℒ_i, the matrix of right eigenvectors is invertible for q > 0 where q ≠ 1 (all possible values q can take), and therefore that all the eigenvectors are linearly independent, which implies diagonalizability for ℒ_i. Below we show 𝒫_2 is invertible. We list all the other cases in <ref>.

    𝒫_2 = [    1 -q^4;    1    1 ]

It follows that (𝒫_2) = q^4+1 ≠ 0 for q > 0. Therefore, 𝒫_2 is invertible for q > 0, and therefore ℒ_2 is diagonalizable for q > 0.

To prove part c, we show that L_Q^D is the direct sum of the diagonalization over all of the communicating classes.

    𝒫𝒜𝒫^-1⊕𝒵


    = (𝒫_9 ⊕⊕_i=1^4 𝒫_6 ⊕⊕_i=1^4 𝒫_4 ⊕⊕_i=1^4 𝒫_3 ⊕⊕_i=1^8 𝒫_2)(𝒜_9 ⊕⊕_i=1^4 𝒜_6 ⊕⊕_i=1^4 𝒜_4 ⊕⊕_i=1^4 𝒜_3 ⊕⊕_i=1^8 𝒜_2)


    (𝒫^-1_9 ⊕⊕_i=1^4 𝒫^-1_6 ⊕⊕_i=1^4 𝒫^-1_4 ⊕⊕_i=1^4 𝒫^-1_3 ⊕⊕_i=1^8 𝒫^-1_2) ⊕𝒵


    = 𝒫_9 𝒜_9 𝒫^-1_9 ⊕⊕_i=1^4 𝒫_6 𝒜_6 𝒫^-1_6 ⊕⊕_i=1^4 𝒫_4 𝒜_4 𝒫^-1_4 ⊕⊕_i=1^4 𝒫_3 𝒜_3 𝒫^-1_3 ⊕⊕_i=1^8 𝒫_2 𝒜_2 𝒫^-1_2 ⊕𝒵


    = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕𝒵


    = L_Q^D.





*

    
    Our goal is to prove that L_Q^D 𝒟 = 𝒟(L_Q^D)^T, and we use <ref>. 

    Consider,
    
    L_Q^D 𝒟 = (𝒫𝒜𝒫^-1⊕𝒵)(𝒫𝒫^T ⊕𝒵)

    
    = (𝒫𝒜𝒫^-1𝒫𝒫^T) ⊕𝒵𝒵 = 𝒫𝒜𝒫^T ⊕𝒵

    
    = 𝒫(𝒫^T (𝒫^T)^-1)𝒜𝒫^T ⊕𝒵

    
    = (𝒫𝒫^T)((𝒫^-1)^T𝒜𝒫^T) ⊕𝒵

    
    = (𝒫𝒫^T)(𝒫𝒜𝒫^-1)^T ⊕𝒵

    
    = (𝒫𝒫^T ⊕𝒵)((𝒫𝒜𝒫^-1)^T ⊕𝒵)

    
    = (𝒫𝒫^T ⊕𝒵)((𝒫𝒜𝒫^-1) ⊕𝒵)^T

    
    = 𝒟(L_Q^D)^T.

    Note that 
    𝒟 = 𝒫𝒫^T ⊕𝒵 = 𝒫_9(𝒫_9)^T ⊕⊕_i=1^4 𝒫_6(𝒫_6)^T ⊕⊕_i=1^4 𝒫_4(𝒫_4)^T ⊕⊕_i=1^4 𝒫_3(𝒫_3)^T ⊕⊕_i=1^8 𝒫_2(𝒫_2)^T ⊕𝒵

    and that
    
    𝒫_2(𝒫_2)^T = [ q^8+1 1-q^4; 1-q^4     2 ].

    contains non-trivial values, completing the proof.

Note that the remaining entries of 𝒟 can be easily calculated from <ref>, and that 𝒫_4(𝒫_4)^T is displayed in <ref>.



§ ALGEBRAIC PROOFS
 



 §.§ W is an irreducible representation


We begin by proving Proposition <ref>. First, we must first establish a few definitions. 

    We define P to be the weight lattice, with λ∈ P called a weight. A vector v_λ is called a highest weight vector if E_1v_λ = E_2 v_λ = E_3v_λ = 0. Then, a _q(_6)-module M is a highest weight module if M=_q(_6)v_λ, and will be denoted M=V(λ).

Since _q(_6) can be triangularly decomposed into its three subalgebras generated by {E_i}, {q^H_i}, and {F_i} (sometimes known as the Cartan decomposition), M can be computed by multiplying compositions of F_is with a highest weight vector v_λ  as described in <cit.>. Note that, in the fundamental representation of _q(_6), each E_i has an empty first column and thus e_1 ∈^6 is a highest weight vector. Denote λ = λ_1L_1 + λ_2L_2 + λ_3L_3. Since e_1 has weight L_1, we can use the following equation from <cit.> to determine the dimension of V(L_1):

    dim(V(λ)) = 1/12∏_1 ≤ i < j ≤ 3 (λ_i - λ_j +j-i)(λ_i+λ_j+6-i-j).

Therefore V(L_1) is a 6-dimensional _q(_6)-module, and V(L_1) is thus the entire fundamental representation.

We will prove multiple statements partially using the theory of crystal bases as stated in <cit.>. We use the correspondence between a weight module V(λ) and its crystal base (ℒ(λ), ℬ(λ)), formed of the crystal lattice and crystal basis respectively. We also use the tensor product rule, Theorem 4.4.1 on pg 83 of <cit.>. 
   V(L_1) ⊗ V(L_1) ≅ V(2L_1) ⊕ V(L_1 + L_2) ⊕ V(0).


    First, define 1 = e_1, 2 = e_2, 3 = e_3, 1̅ = e_4, 2̅ = -e_5, 3̅ = e_6. Then, note that V(L_1) has crystal base as shown in Figure <ref>, with arrow index i corresponding to Kashiwara operator f_i. 

    

    Then, by the tensor product rule, we can draw the crystal graph of V ⊗ V as shown in Figure <ref>. 




    Thus, by reading off the disjoint connected components in the crystal graph of Figure <ref>, we see that
    
    V(L_1) ⊗ V(L_1) ≅ V(2L_1) ⊕ V(L_1 + L_2) ⊕ V(0).

    Note that, by applying the corresponding F_i, we can trace a path of arrows to each vertex in W and obtain the corresponding basis vector of W. This process is used to generate the compositions of F_is shown in Lemma <ref>.


Now, we move on to constructing V(2L_1) explicitly.
 The _q(_6)-module V(2L_1) is 20-dimensional with basis

    { e_1 ⊗ e_1   ,  e_1 ⊗ e_2 + q^-1e_2 ⊗ e_1, e_1⊗ e_3 + q^-1e_3 ⊗ e_1, 
       -e_1 ⊗ e_4 +q^-1e_2 ⊗ e_5 - q^-2e_4 ⊗ e_1+q^-1e_5 ⊗ e_2, e_1⊗ e_5 +q^-1e_5 ⊗ e_1, 
       - e_1 ⊗ e_6-q^-1e_6 ⊗ e_1,  (q+q^-1)e_2 ⊗ e_2,e_2 ⊗ e_3 + q^-1 e_3 ⊗ e_2, 
       -e_2 ⊗ e_4 -q^-1e_4 ⊗ e_2, e_2 ⊗ e_5 - q^-1e_3 ⊗ e_6+q^-2e_5 ⊗ e_2 -(q^2+1)e_6 ⊗ e_3,
       -e_2 ⊗ e_6 -q^_1e_6⊗ e_2,(q^-1+q)e_3 ⊗ e_3, 
       -(q^2 +2 +q^-2)e_3 ⊗ e_4 - (q+2q^-1+q^-3)e_4 ⊗ e_3, 
       (q^2 + 2+q^-2)e_3 ⊗ e_5 + (q+2q^-1+q^-3)e_5 ⊗ e_3,q^-2e_3 ⊗ e_6 -q^-1e_6 ⊗ e_3, 
       (q^4 + 4q^2 + 6 + 4q^-2 + q^-4)e_4 ⊗ e_4, 
       -(q^2+3+3q^-2+q^-4)e_4 ⊗ e_5 - (q^3 +3q+3q^-1+q^-3)e_5 ⊗ e_4  
       (q^-2+1)e_4 ⊗ e_6 +(q+q^-1)e_6 ⊗ e_4,(q^2+2+q^-2)e_5 ⊗ e_5, 
       -(q^-2+1)e_5 ⊗ e_6 - (q^-1+q)e_6 ⊗ e_5, (q^-1+q)e_6 ⊗ e_6}



    Define π_V(F_i) to be the projection of Δ(F_i) into V = ^6 ⊗^6. The tensor product assumes the role of the Kronecker product in this 36-dimensional representation V of _q(_6). Note that {π_V(E_i)} annihilates e_1 ⊗ e_1, making e_1 ⊗ e_1 a highest weight vector with highest weight 2L_1. To generate V(2L_1), we use Lemma <ref> to obtain the correct compositions of F_is applied to e_1 ⊗ e_1 to form each basis vector: we apply F_is corresponding to the arrows in the path from e_1 ⊗ e_1 to each e_i ⊗ e_j contained in the upper right-hand cycle as displayed Figure <ref>. A few examples are done below, and the rest are left to the reader. The notation F_ijk is used to shorten F_iF_jF_k.
    
        π_V(F_1)(e_1 ⊗e_1)    = (1 ⊗F_1)(e_1 ⊗e_1) + (F_1 ⊗q^-H_1)(e_1 ⊗e_1) 

           = e_1 ⊗e_2 + q^-1e_2 ⊗e_1 

        π_V(F_11) π_V(F_1F_1)(e_1 ⊗e_1)    = (q+q^-1)e_2 ⊗e_2  

        π_V(F_11332211) (e_1 ⊗e_1)    = (q^4 + 4q^2 + 6 + 4q^-2 + q^-4)e_4 ⊗e_4 
    
    After complete evaluation, we obtain the basis shown in Equation <ref> for V(2L_1). Note that the number of basis vectors generated agrees with the dimension formula in Equation <ref>.

Using Lemma <ref> and Lemma <ref>, we are now ready to prove Proposition <ref>.
*

    Note that the basis vectors of V(2L_1) span 
    W and are linearly independent. Since finite-dimensional highest weight modules of _q(_6) are irreducible representations, W is an irreducible representation.

As a sanity check, if C is a central element of _q(_6), we will see that π_V(C)|_W is a constant times Id_20 × 20.

    In 2020, Kuan, Landry, Lin, Park, and Zhou <cit.> found the following central element of _q(_6) (note that the fundamental representation of _2n presented in that paper is slightly incorrect, leading to some sign errors in its central elements <cit.>. The E_3 used in that paper is -E_3 under our notation). We write F_ij to shorten F_iF_j and r to represent q+q^-1. 
    
    C    = q^-4-2H_1-H_2-H_3
                    +q^-2-H_2-H_3
                    +q^H_2-H_3
                    +q^H_3-H_2
                    +q^2+H_2+H_3
       +q^4+2H_1+H_2+H_3
                    +r^2/q^3F_1q^-H_1-H_2-H_3E_1
                    +r^2/qF_2q^-H_3E_2 
       +r^2/qF_3q^-H_2E_3
                    +r^2qF_2q^H_3E_2
                    +r^2qF_3q^H_2E_3
                    +r^2q^3F_1q^H_1+H_2+H_3E_1 
       +r^2/q^3(qF_12-F_21)q^-H_1-H_3(qE_21-E_12) 
       +r^2/q^3(qF_13-F_31)q^-H_1-H_2(qE_31-E_13) 
       +r^2q(qF_21-F_12)q^H_1+H_3(qE_12-E_21) 
       +r^2q(qF_31-F_13)q^H_1+H_2(qE_13-E_31) 
       +r^2/q^3(q^2F_123-qF_213-qF_312+F_231)q^-H_1(q^2E_231-qE_312-qE_213+E_123) 
       +r^2/q(q^2F_231-qF_312-qF_213+F_123)q^H_1(q^2E_123-qE_213-qE_312+E_231) 
       +r^4/q^2((q^2+1)F_1231-qF_1312-qF_2131)((q^2+1)E_1231-qE_1312-qE_2131) 
       +r^4F_2F_3E_2E_3

    By some tedious 36× 36 matrix computations, one can find π_V(C). Then, construct the change of basis matrix Y with first 20 columns being the above-constructed basis of W and the last 16 columns extending to a basis of V. Then, the matrix Y^-1π_V(C)Y has an upper left 20 × 20 block of (q^8+q^2+2+q^-2+q^-8) Id_20 × 20 = π_V(C)|_W.




 §.§ Decompositions of W tensor W

In this section, we analyze the structure of W ⊗ W. This will allow us to deduce properties of π_W ⊗ W(C) in Section <ref>.  


  §.§.§ Decomposition of W tensor W into irreducible representations

In order to decompose W ⊗ W into a direct sum of irreducible representations, we would like to use some analog of the tensor product rule as done in Lemma <ref>. However, since our current crystal graph of W is three-dimensional, creating the crystal graph of W ⊗ W is very difficult. Therefore, we introduce Young tableaux to be able to visualize W ⊗ W, and use the corresponding tensor product rule to decompose W ⊗ W. An interested reader may reference <cit.> chapters 7 and 8 for all appropriate background.

The chain of correspondence is as follows: ℬ(2L_1) describes W, as seen in Lemma <ref>, and

    ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
        ) ℬ(𝒴)

describes ℬ(2L_1). Thus, the decomposition of ℬ(𝒴) ⊗ℬ(𝒴) dictates the structure of W ⊗ W. This brings us to Lemma <ref>. 
  W ⊗ W decomposes as follows into irreducible representations:
    
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0).

    As expected, the sum of the dimensions of these irreducible representations is 400 = dim(W⊗ W).


    As can be read off of Figure <ref>, 
    
    ℬ(2L_1) = { (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
            [circle] (ablah) at (.25,-.05) a;
            [circle] (ablah) at (.75,0) b;
        |  b ≽ a }

    We choose the English notation and convention of reading Young diagrams down columns from right to left. By the tensor product rule for Young diagrams (Theorem 8.6.6, pg 206 of <cit.>),
    
    ℬ(𝒴) ⊗ℬ(𝒴) ≅⊕_b_1 ⊗ b_2 ∈ Wℬ(𝒴[b_1, b_2]).

    To compute the right hand side, note that:
    
        ℬ(𝒴[v_1, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1.5, .25);
        (40) at (2, -.25);
        (41) at (2, .25);
        [thick, -] (00) – (40) – (41) – (01) – (00);
        [thick, -] (10) – (11);
        [thick, -] (20) – (21);
        [thick, -] (30) – (31);
    
    ) 

    ℬ(𝒴[v_2, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1.5, .25);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (00) – (30) – (31) – (01) – (00);
        [thick, -] (10) – (11);
        [thick, -] (20) – (21);
        [thick, -] (02) – (12);
        [thick, -] (01) – (02);
        [thick, -] (11) – (12);
    
    ) 

    ℬ(𝒴[v_2, v_2])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1, -.75);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (01) – (02) – (31) – (21) – (01) ;
        [thick, -] (00) – (20);
        [thick, -] (11) – (12);
    
    ) 

    ℬ(𝒴[v_1̅, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        [thick, -] (00) – (20) – (21) – (01) – (00);
        [thick, -] (10) – (11);
    
    ) 

    ℬ(𝒴[v_1̅,v_2])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1, -.75);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (01) – (02) – (12) – (11) – (01) ;
        [thick, -] (00) – (10);
    
    ) 

    ℬ(𝒴[v_1̅, v_1̅])    = ℬ(∅). 
    
    Thus, we have that
    
    ℬ(𝒴) ⊗ℬ(𝒴) ≅ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1.5, .25);
             (40) at (2, -.25);
             (41) at (2, .25);
            [thick, -] (00) – (40) – (41) – (01) – (00);
            [thick, -] (10) – (11);
            [thick, -] (20) – (21);
            [thick, -] (30) – (31);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1.5, .25);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (00) – (30) – (31) – (01) – (00);
            [thick, -] (10) – (11);
            [thick, -] (20) – (21);
            [thick, -] (02) – (12);
            [thick, -] (01) – (02);
            [thick, -] (11) – (12);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1, -.75);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (01) – (02) – (31) – (21) – (01) ;
            [thick, -] (00) – (20);
            [thick, -] (11) – (12);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1, -.75);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (01) – (02) – (12) – (11) – (01) ;
            [thick, -] (00) – (10);
        ) ⊕ℬ(∅)

    which implies that
     
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0).

    As for the equality of dimensions, note that
    
        dim(W ⊗W)    = 400 
 
           = 105+175+84+20+15+1 

           = ∑_λdimV(λ)
    
    using the dimension formula Equation <ref>. The order of dimensions in the above sum is the same order in which the irreducible representations are listed in the decomposition above.

We proceed to investigating our next trait of W ⊗ W.



  §.§.§ Decomposition of W tensor W into weight spaces

By definition, each highest-weight _q(_6)-module has a weight space decomposition. Therefore, we can express W ⊗ W as a direct sum of weight spaces using the Young tableaux found in Lemma <ref>. We are most interested in the dimension of each weight space, which will help deduce the shape of π_W ⊗ W(C). 


    W ⊗ W admits a decomposition into eighty-five weight spaces, one with dimension 22, twelve with dimension 12, six with dimension 8, twenty-four with dimension 4, twelve with dimension 3, twenty-four with dimension 2, and six with dimension 1.


    For ease of notation, define
    
    L_-i -L_i

    For 1 ≤ i ≤ 3. Recall that the weights associated with the standard basis vectors of the fundamental representation ^6 of _6 are as follows:
    
    
    It is a property of weights that if v and v' are vectors in a representation V with respective weights λ and λ', then v ⊗ v' is a vector in V ⊗ V with weight λ + λ'. Thus, we can observe from the crystal graph of W that the weights of basis vectors in W take the following forms, for i,j ∈{1,2,3,-1,-2,-3} with i,-i,j,-j all unique:
    
    {0,2L_i,L_i+L_j}

    Note that two basis vectors in W (e_1 ⊗ e_4 and e_2 ⊗ e_5) have weight 0, and each of the other eighteen basis vectors in W has a unique weight corresponding to one of the forms above. By considering possible sums of the weights above, the weights of vectors in W ⊗ W take one of the following forms, for i,j,k ∈{1,2,3,-1,-2,-3} with i,-i,j,-j,k,-k all unique:
    
    {0,L_i + L_j,2L_i,2L_i + L_j + L_k,2L_i + 2L_j,3L_i + L_j,4L_i}

    Now we count the number of weight spaces corresponding to each of the forms above.
    
        
  * There is evidently one weight space with weight 0.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since the pairs (i,j) and (j,i) yield the same weight, there are twelve weight spaces with a weight of the form L_i + L_j.
        
  * Since there are 6 possible values for i, there are six weight spaces with a weight of the form 2L_i.
        
  * There are 6 · 4 · 2 = 48 possible triplets (i,j,k). Since the triplets (i,j,k) and (i,k,j) yield the same weight, there are twenty-four weight spaces with a weight of the form 2L_i + L_j + L_k.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since the pairs (i,j) and (j,i) yield the same weight, there are twelve weight spaces with a weight of the form 2L_i + 2L_j.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since each pair yields a distinct weight, there are twenty-four weight spaces with a weight of the form 3L_i + L_j.
        
  * Since there are 6 possible values for i, there are six weight spaces with a weight of the form 4L_i.
    
    Now we count the dimension of each type of weight space. We can do so by counting the number of basis vectors w_1 ⊗ w_2 ∈ W ⊗ W such that the sum of the weights corresponding to w_1 and w_2 in W equals the desired weight. Denote the weights in W corresponding to w_1 and w_2 by λ_1 and λ_2, respectively.
    
        
  * If λ_1+λ_2 = 0, then either
        
            
  * λ_1 = 0. Then λ_2 = 0 as well. There are thus two options for each of w_1 and w_2, and thus four total possibilities in this case.
            
  * λ_1  0. Then there are eighteen possible values for w_1 with nonzero weights, and exactly one possible value for w_2 such that λ_2 = -λ_1.
        
        Thus, the dimension of the weight space of W ⊗ W with weight 0 is twenty-two.
        
  * If λ_1+λ_2 takes the form L_i+L_j, then either
        
            
  * λ_1 = L_i + L_j and λ_2 = 0; or λ_1 = 0 and λ_2 = L_i + L_j. Since the weight space of W with weight 0 has dimension 2, there are four total possibilities in this case.
            
  * λ_1 = L_i - L_j and λ_2 = 2L_j; or λ_1 = 2L_j and λ_2 = L_i - L_j; or λ_1 = -L_i + L_j and λ_2 = 2L_i; or λ_1 = 2L_i and λ_2 = -L_i + L_j. There are four total possibilities in this case.
            
  * λ_1 = L_i + L_k and λ_2 = -L_j - L_k; or λ_1 = -L_j - L_k and λ_2 = L_i + L_k. Since there are two possibilities for k once i and j are fixed, there are four total possibilities in this case.
        
        Thus, each weight space of W ⊗ W with a weight of the form L_i + L_j has dimension twelve.
        
  * If λ_1 + λ_2 takes the form 2L_i, then either
        
            
  * λ_1 = 2L_i and λ_2 = 0; or λ_1 = 0 and λ_2 = 2L_i. Since the weight space of W with weight 0 has dimension 2, there are four total possibilities in this case.
            
  * λ_1 = L_i + L_j and λ_2 = L_i - L_j. Since there are four possibilities for j  once i is fixed, there are four total possibilities in this case.
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i has dimension eight.
        
  * If λ_1 + λ_2 takes the form 2L_i + L_j + L_k, then either
        
            
  * λ_1 = 2L_i and λ_2 = L_j + L_k
            
  * λ_1 = L_j + L_k and λ_2 = 2L_i
            
  * λ_1 = L_i + L_j and λ_2 = L_i + L_k
            
  * λ_1 = L_i + L_k and λ_2 = L_i + L_j
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i + L_j + L_k has dimension four.
        
  * If λ_1 + λ_2 takes the form 2L_i + 2L_j, then either
        
            
  * λ_1 = 2L_i and λ_2 = 2L_j
            
  * λ_1 = 2L_j and λ_2 = 2L_i
            
  * λ_1 = L_i + L_j and λ_2 = L_i + L_j
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i + 2L_j has dimension three.
        
  * If λ_1 + λ_2 takes the form 3L_i + L_j, then either
        
            
  * λ_1 = 2L_i and λ_2 = L_i + L_j
            
  * λ_1 = L_i + L_j and λ_2 = 2L_i
        
        Thus, each weight space of W ⊗ W with a weight of the form 3L_i + L_j has dimension two.
        
  * If λ_1 + λ_2 takes the form 4L_i, then λ_1 = 2L_i and λ_2 = 2L_i, so each weight space of W ⊗ W with a weight of the form 4L_i has dimension one.
    
    



    This is a direct consequence of Lemma <ref> and Lemma <ref>.




 §.§ Block form of pi(W tensor W)(C)
 




  §.§.§ Block sizes of pi(W tensor W)(C)

Since W ⊗ W can be decomposed into a direct sum of weight spaces, any matrix in W ⊗ W can be expressed as a direct sum of matrices in the weight spaces of W ⊗ W. This yields Proposition <ref>, restated as the following corollary of Lemma <ref>:

    π_W ⊗ W(C) admits a block diagonal decomposition into a direct sum of one 22 × 22 block, twelve 12 × 12 blocks, six 8 × 8 blocks, twenty-four 4 × 4 blocks, twelve 3 × 3 blocks, twenty-four 2 × 2 blocks, and six 1 × 1 blocks. The basis in respect to which results in π_W ⊗ W(C) blocked this way is included in Section <ref>.



Note that since _6 and 𝔰𝔩_4 are isomorphic as Lie algebras, computing the Kostka numbers for 𝔰𝔩_4 would also yield the block sizes found above. For example, the 22× 22 block can be calculated by 22=3+6+7+2+3+1 where each summand appears as K_λμ for the following choices of λ and μ.

    K_λμ = 7,      λ = (4,3,1,0), μ = (2,2,2,2)
    
    K_λμ = 6,      λ = (4,2,2,0), μ = (2,2,2,2) 
    
    K_λμ = 3,      λ = (4,4,0,0), μ = (2,2,2,2) 
    
    K_λμ = 2,      λ = (2,2,0,0), μ = (1,1,1,1) 
    
    K_λμ = 3,      λ = (2,1,1,0), μ = (1,1,1,1) 
    
    K_λμ = 1,      λ = (0,0,0,0), μ = (0,0,0,0)





  §.§.§ Entries of pi(W tensor W)(C)

Since the coproduct Δ is a homomorphism, π_^6 ⊗^6 ⊗^6 ⊗^6(C) can be computed from Definition <ref> by computing Δ^3(E_i), Δ^3(F_i), and Δ^3(q^H_i) for 1 ≤ i ≤ 3 and substituting them for E_i, F_i, and q^H_i, respectively, in the equation in Example <ref>. By Definition <ref>, for all 1 ≤ i ≤ 3,

    Δ^3 (E_i) = E_i ⊗ 1 ⊗ 1 ⊗ 1 + q^H_i⊗ E_i ⊗ 1 ⊗ 1 + q^H_1⊗ q^H_1⊗ E_i ⊗ 1 + q^H_1⊗ q^H_1⊗ q^H_1⊗ E_1 
       Δ^3 (F_i) = 1 ⊗ 1 ⊗ 1 ⊗ F_i + 1 ⊗ 1 ⊗ F_i ⊗ q^-H_i + 1 ⊗ F_i ⊗ q^-H_i⊗ q^-H_i + F_i ⊗ q^-H_i⊗ q^-H_i⊗ q^-H_i
       Δ^3 (q^H_i) = q^H_i⊗ q^H_i⊗ q^H_i⊗ q^H_i

Denote the matrix resulting from the above procedure by π_^1296 (C).


Denote the basis of W established in the proof of Lemma <ref> by B_W  (w_1,w_2,…,w_20). Recall the 36 × 36 change of basis matrix Y defined in Example <ref>, and note that a change of basis by Y maps w_i to the standard basis vector e_i of ^6 ⊗^6 for 1 ≤ i ≤ 20. Thus, a change of basis by Y ⊗ Y in ^6 ⊗^6 ⊗^6 ⊗^6 maps B_W ⊗ B_W to {e_i ⊗ e_j : 1 ≤ i,j ≤ 20}. Since for any 1 ≤ i,j ≤ 36, e_i ⊗ e_j is the (36(i-1)+j)th basis vector of ^6 ⊗^6 ⊗^6 ⊗^6, π_W ⊗ W(C) = π_^1296(C) |_W ⊗ W can be found by taking the (36(i-1)+j)th row and column of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y) for all 1 ≤ i,j ≤ 20. Since the e_i ⊗ e_jth row or column of π_W ⊗ W(C) corresponds directly to the (36(i-1)+j)th row or column of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y), we only need to determine a few of the values of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y) to compute some of the blocks of π_W ⊗ W(C). Explicit values of π_W ⊗ W(E_i),π_W ⊗ W(F_i),π_W ⊗ W(q^H_i) for 1 ≤ i ≤ 3 are given in Section <ref>



  §.§.§ Eigenvalues of pi(W tensor W)(C)

Given W ⊗ W's decomposition into highest weight _q(_6)-modules, we are also able to deduce the eigenvalues of π_W ⊗ W(C).

    π_W ⊗ W(C) has eigenvalues of
    
           q^12+q^2+2+q^-2+q^-12, 

           q^10+q^4 + 2 + q^-4+q^-10, 

           q^8 +q^6+2+q^-6+q^-8, 

           q^8+q^2+2+q^-2+q^-8 

           q^6 + q^4 +2 + q^-4+q^-6, 

           q^4+q^2+2+q^-2+q^-4
    
    with respective multiplicities of 105, 175, 84, 20, 15, 1.


     By Lemma <ref>,
     
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0)

    with respective dimensions of 105, 175, 84, 20, 15, 1. Since each of these representations is irreducible, 
    π_W ⊗ W(C)|_V(λ) = cId
 for some c ∈[q,q^-1]. Also, by construction of each V(λ), v_λ is a basis vector of V(λ). Therefore, in changing the basis of W ⊗ W to compute π_W ⊗ W(C)|_V(λ), we will have 
    
    π_W ⊗ W(C)|_V(λ)(1,1) = ⟨ v_λ|π_W ⊗ W(C)|_V(λ)|v_λ⟩

    since we may express each highest weight vector v_λ as being a 400-dimensional vector with a 1 in the i^th position and 0s elsewhere. Thus, to find the c associated with each v_λ, we must only find the i^th diagonal entry of π_W ⊗ W(C). We will then have that c is an eigenvalue of multiplicity dim(V(λ)) of this matrix with changed basis. Since a change of basis does not impact the eigenvalues of the matrix and π_W ⊗ W(C) is a block matrix by Lemma <ref>, c is an eigenvalue of π_W ⊗ W(C) with multiplicity dim(V(λ)). 
    Next, as v_λ is a highest weight vector of weight λ, by definition,
    
    E_1v_λ = E_2v_λ = E_3v_λ = 0.

    Thus, 
    
    π_W ⊗ W(C)v_λ = π_W ⊗ W(q^-4-2H_1-H_2-H_3
                +q^-2-H_2-H_3
                +q^H_2-H_3
                +q^H_3-H_2
                +q^2+H_2+H_3 
                +q^4+2H_1+H_2+H_3)v_λ.

    Since each q^H_i is a diagonal matrix, we may use Section <ref> to find the corresponding diagonal entry to each highest weight vector in a feasible way computationally. For example, v_4L_1 = e_1^⊗ 2 where e_1 is 20-dimensional, and thus π_W ⊗ W(C) has an eigenvalue of 
    
    c    =  ⟨e_1^⊗2|π_W ⊗W(C)|e_1^⊗2 ⟩

       =⟨e_1^⊗2|π_W ⊗W(q^-4-2H_1-H_2-H_3
            +q^-2-H_2-H_3
            +q^H_2-H_3
            +q^H_3-H_2
            +q^2+H_2+H_3 
            +q^4+2H_1+H_2+H_3)| e_1^⊗2⟩

               = q^-4·q^-4 ·q^-4 ·1 ·1 + q^-2·1 ·1+1·1+1·1+q^2·1·1 +q^4·q^4·q^4·1·1 

           = q^12+q^2+2+q^-2+q^-12
    
    with multiplicity dim(V(4L_1)) = 105. The remaining eigenvalues can be found similarly.
    



    Recall that _6 is a rank 3 Lie algebra which is isomorphic to the rank 3 Lie algebra 𝔰𝔩_4. In this case, previous research in <cit.> allows us to express C using generators from both _6 and 𝔰𝔩_4 in terms of the above eigenvalues. We choose not to explore this, but someone who wishes to relate <cit.>'s central element C to the Harish Chandra isomorphism may reference <cit.>.    




 §.§ Type A Ground State Transformation

Following the path of previous research, we used Section <ref> as a starting point to form our ground state transformation. We perform a ground state transformation so that the resulting matrix has rows which sum to 0, allowing us to use this transformed matrix as a generator for a Markov process. We will then compare this generator to that of the probability section. 


    
    A ground state transformation is a map from a Hamiltonian matrix, H, to a matrix whose rows sum to 0. We define such a map to be of the following form, where G is a diagonal matrix and a is in the underlying field.
    
    G^-1HG - a Id

    We call G a ground state transformation matrix of H.


Section <ref> provides a ground state transformation that worked with Type A_n Lie algebras. However, as can be seen using the respective crystal graphs, Type D_n Lie algebras have more complicated behavior: the crystal graph for a fundamental representation of _q(𝔰𝔩_4) is shown in Figure <ref>.


As a result, the crystal graph for V ⊗ V of _q(𝔰𝔩_4) is the graph shown in Figure <ref>. Note that the cycle from the highest weight vector to the lowest weight vector is straightforward.
 

Instead, in _q(_6), the crystal graph for V ⊗ V shown in Figure <ref> was much less simple due to the more complex crystal graph for V shown in Figure <ref>. In this case, the shortest path from the highest weight vector to the lowest weight vector requires eight applications of F_is as shown in the third example within the proof of Lemma <ref>.

Thus, intuitively, when we apply the tensor product rule again to W, the path from the highest weight vector to the lowest weight vector should be even more intricate. Therefore, the ground state transformation in Section <ref> should include more complicated structures than solely permutations of F_1^k_1F_2^k_2F_3^k_3 in order to generate a complete matrix G. This result is formalized below.
*

    By the crystal graph for V shown in Figure <ref>, which is based on the fundamental representation defined in Definition <ref>, we see that we will not be able to compute e_4 without applying either F_1F_2F_3F_1 or F_1F_3F_2F_1.
    
Now, note that W is a subspace of Sym^2_q(^6); in particular, as shown in Lemma <ref>, W has basis elements of the form:

    { p_i(q)   e_i ⊗e_i, p_i(q)e_i ⊗e_j+p_j(q)e_j ⊗e_i, |i ≠j, (i,j) ≠(1,4),(2,5),(3,6), p_j  Laurent polynomial} ∪

       { -e_1 ⊗e_4 +q^-1e_2 ⊗e_5 - q^-2e_4 ⊗e_1+q^-1e_5 ⊗e_2, e_2 ⊗e_5 - q^-1e_3 ⊗e_6+q^-2e_5 ⊗e_2 -(q^2+1)e_6 ⊗e_3}

Thus, applying a sequence of F_is to obtain, for example, 

    ⟨ e_1 ⊗ e_2| π_W(Δ(F_1))| e_1 ⊗ e_1 ⟩ = 1

will also yield

    ⟨ e_2 ⊗ e_1| π_W(Δ(F_1))| e_1 ⊗ e_1 ⟩ = q^-1

because e_1 ⊗ e_2+qe_2 ⊗ e_1 is a basis vector of W.

However, the composition of F_is required to obtain e_4 ⊗ e_4, Π_i (Δ F_i), will only produce a nonzero 

    ⟨ e_4 ⊗ e_4| π_W(Π_i (Δ(F_i)))| e_1 ⊗ e_1 ⟩

and result in 0 when taking, for any other (i,j) ≠ (4,4),

    ⟨ e_i ⊗ e_j| π_W(Π_i (Δ(F_i)))| e_1 ⊗ e_1 ⟩

as e_4 ⊗ e_4 does not appear in any other basis vector of W. Therefore, since e_4 ⊗ e_4 lies in the bottom right corner of Figure <ref>, it is clear that in order to obtain a nonzero value corresponding to e_4 ⊗ e_4, we must follow the arrows and apply a composition of the form

    ⟨ e_4 ⊗ e_4| π_W(Δ(F_1) ·Π_i (Δ(F_i)) ·Δ(F_1))| e_1 ⊗ e_1 ⟩

with, in particular, 

    ⟨ e_4 ⊗ e_4| π_W(Δ(F_1)Δ(F_1)Δ(F_3)Δ(F_3)Δ(F_2)Δ(F_2)Δ(F_1)  Δ(F_1))| e_1 ⊗ e_1 ⟩ = q^4 + 4q^2 + 6 + 4q^-2 + q^-4.

Finally, since W is a subset of a symmetric tensor, W ⊗ W will be as well. Therefore, we must again find a composition Π_i(Δ (F_i)) that produces exactly e_4^⊗ 4, and it will not be enough to produce any other vector. By the tensor product rule, this will yet again result in a nonzero value to an expression only in the following structure

    ⟨ e_4^⊗ 4| π_W(Δ(F_1) ·Π_i (Δ(F_i)) ·Δ(F_1))| e_1^⊗ 4⟩





 §.§ Finding a Markov Process Generator




  §.§.§ Partial Type A Ground State Transformation

 
    We define a partial Type A ground state transformation of π_W ⊗ W(C) to be a map
    
    π_W ⊗ W(C) ↦. ( G^-1π_W ⊗ W(C)G - aId) | _K

    where K is the vector subspace of W ⊗ W which is the span of all basis vectors u_i ⊗ u_j for which there exists a triple (k_1,k_2,k_3) of nonnegative integers satisfying
    
    ⟨ u_i ⊗ u_j | ∏_i=1^3 Δ(F_ℓ_i)^k_ℓ_i| e_1 ⊗ e_1 ⟩ 0

    where (ℓ_1,ℓ_2,ℓ_3) is some permutation of (1,2,3), and where G is a 400 × 400 diagonal matrix such that the diagonal entry of G corresponding to u_i ⊗ u_j is the unique nonzero value above.

Note that by Lemma <ref>, the partial Type A ground state transformation is not unique.



  §.§.§ Selecting a Ground State Transformation

We now extend the partial Type A ground state transformation to a ground state transformation of π_W ⊗ W(C). Define a  q^12 + q^2 + q^-2 + q^-12.




    Let G be a ground state transformation matrix for π_W ⊗ W(C), i.e., a diagonal matrix such that each row of a^-1G^-1π_W ⊗ W(C)G - Id sums to 0, and let g_i denote the ith diagonal entry of G for 1 ≤ i ≤ 400. Then the vector
    
    g⃗[   g_1;   g_2;     ⋮; g_400 ]

    satisfies π_W ⊗ W(C) g⃗ = ag⃗. This condition is both sufficient and necessary, i.e., if g⃗ is an eigenvector of π_W ⊗ W(G) with eigenvalue a, the diagonal matrix G with entries corresponding to g⃗ as above is such that the rows of a^-1G^-1π_W ⊗ WG - Id sum to 0.



    Define the matrix L by
    
    L  a^-1G^-1π_W ⊗ W(C)G - Id

    By definition, each row of L sums to 0. Then we have
    
    G^-1(π_W ⊗ W(C))G - aId   = aL 
    
            G^-1(π_W ⊗ W(C))G    = aL + aId
    π_W ⊗ W(C)G    = aGL + aG

    Since G is diagonal and each row of L sums to 0, each row of aGL sums to 0. Thus, the sum of the elements in each row of π_W ⊗ W(C) G equals the sum of the elements in the same row of aG. Defining g_i as above, we can write the sum of the ith row of π_W ⊗ W(C)G as ∑_j=1^400π_W ⊗ W(C)_i,j g_j, and the sum of the ith row of aG as ag_i. Thus, for all 1 ≤ i ≤ 400,
    
    ∑_j=1^400π_W ⊗ W(C)_i,j g_j    = ag_i

    This is equivalent to saying that π_W ⊗ W(C) g⃗ = ag⃗.
    
    
    Reversing the steps above shows that the condition π_W ⊗ W(C)g⃗ = ag⃗ is sufficient as well as necessary.


Thus, any partial Type A ground state transformation matrix G of π_W ⊗ W(C) which satisfies the condition in Lemma <ref> is a ground state transformation matrix of π_W ⊗ W(C). Finding diagonal entries of G as described in Definition <ref>  and solving the condition in Lemma <ref> as a system of linear equations yields a solution space 𝒮 with 37 unknowns.


    Given any G ∈𝒮,
    
    a^-1G^-1HG - Id

    is a generator for a Markov process.


    Immediate by the definition of 𝒮.

Note that the proof of Theorem <ref> is an immediate consequence of 𝒮 being nonempty. We give one such generator in Section <ref> by setting all unknowns to 0 for ease of computation. One detail to note is that this generator has dimension 196 = 14^2, which suggests that, when interpreted as an interacting particle system, the process allows for 14 different states — all but four particles of the same class — at each site.



 §.§ Difference with Probability Generator

To relate our algebraic findings to those of probability, we end by comparing the two generators. First, note that the 2 × 2 and 3 × 3 blocks of the algebraically-produced Markov generator in Section <ref> are a constant multiple of the 2 × 2 and 3 × 3 blocks of the probabilistically-produced Markov generator in Section <ref>. However, we notice that any generator matrix given by a ground state transformation as described in Definition <ref> will not produce a generator matrix for a Markov process matching the probabilistically-produced generator matrix. We now prove Proposition <ref>.


    We will show that no block of any ground state transformation of π_W ⊗ W(C) can be a constant multiple of the 4× 4 block of L_Q in Section <ref> corresponding to the communicating class

    ( ⟨ 3 , 0⟩, ⟨1 , 2⟩, ⟨2 , 1⟩, ⟨0 , 3⟩).

    Thus, L_Q must be different from any Markov generator produced from π_W ⊗ W(C).

    Note that the above communicating class contains one class 1 particle and one class 2 particle per Section <ref>. As proven in Proposition 2.5 of <cit.>, this communicating class must correspond to the weight 
    
    2 λ_i + λ_j +λ_k

    found in Section <ref>. As discussed in Lemma <ref>, there are 24 weight spaces of dimension 4 associated with 2 λ_i + λ_j +λ_k, and thus 2 λ_i + λ_j +λ_k corresponds to the twenty-four 4 × 4 blocks in π_W ⊗ W. It is therefore sufficient to show that none of the 4 × 4 blocks of π_W ⊗ W(C), which are listed in Section <ref>, can be mapped to the 4 × 4 block in L_Q corresponding to the above communicating class.

    By referencing Section <ref>, we notice that each 4 × 4 block does not contain any zero entries. However, every 4 × 4 block listed in Section <ref> has an off-diagonal entry which takes the value of 0.

    Since conjugation by a diagonal matrix will not change a 0-entry in a matrix and the 0-entries are off-diagonal, Definition <ref> confirms that no ground state transformation of π_W ⊗ W(C) will result in L_Q.


The computationally-heavy verification which involves disregarding the relationship between weights and particles and instead checking 4 × 4 blocks after trying all possible ground state transformations on π_W ⊗ W(C) will yield the same result, as this proof is equivalent to the one detailed above.



§ APPENDIX
 


 §.§ Code for Probabilistic Sections

The SymPy code used to verify the generator matrix L_Q, limits of component generator matrices, stationary distributions, spectral gaps, and matrices of Markov duality can be found at the authors' https://github.com/evaengel/stochastic-fusionGitHub repository. Because the largest communicating classes only have size 9, the matrices are small enough that the calculations can be done by hand. Those details are omitted from this paper, and the SymPy code only serves as an independent verification.



 §.§ List of Communicating Classes by Size
 
The following communicating classes are for the two-γ lattice site Type D ASEP.

The ordered 9-state communicating class follows:

    ( ⟨ 3, 3 ⟩,⟨ 0, 33 ⟩, ⟨ 33,0 ⟩,⟨ 1,32 ⟩,⟨ 32,1 ⟩,⟨ 2,31 ⟩, ⟨ 31,2 ⟩,⟨ 11, 22 ⟩,⟨ 22, 11 ⟩)


The ordered 6-state communicating classes follow:

    ( ⟨ 1 , 3⟩, ⟨3 , 1⟩, ⟨0 , 31⟩, ⟨31 , 0⟩, ⟨2 , 11⟩, ⟨11 , 2 ⟩)


    ( ⟨ 2 , 3⟩, ⟨3 , 2⟩, ⟨0 , 32⟩, ⟨32 , 0⟩, ⟨1 , 22⟩, ⟨22 , 1⟩)


    ( ⟨ 3 , 31⟩, ⟨31 , 3⟩, ⟨1 , 33⟩, ⟨33 , 1⟩, ⟨11 , 32⟩, ⟨32 , 11⟩)


    ( ⟨ 3 , 32⟩, ⟨32 , 3⟩, ⟨2 , 33⟩, ⟨33 , 2⟩, ⟨22 , 31⟩, ⟨31 , 22⟩)


The ordered 4-state communicating classes follow:

    ( ⟨ 3 , 0⟩, ⟨1 , 2⟩, ⟨2 , 1⟩, ⟨0 , 3⟩)


    (⟨ 3 , 11⟩, ⟨1 , 31⟩, ⟨11 , 3⟩, ⟨31 , 1⟩)


    (⟨ 3 , 22⟩, ⟨2 , 32⟩, ⟨22 , 3⟩, ⟨32 , 2⟩)


    (⟨ 33 , 3⟩, ⟨31 , 32⟩, ⟨32 , 31⟩, ⟨3 , 33⟩)


The ordered 3-state communicating classes follow:

    (⟨ 1 , 1⟩, ⟨0 , 11⟩, ⟨11 , 0⟩)


    (⟨ 2 , 2⟩, ⟨0 , 22⟩, ⟨22 , 0⟩)


    (⟨ 31 , 31⟩, ⟨11 , 33⟩, ⟨33 , 11⟩)


    (⟨ 32 , 32⟩, ⟨22 , 33⟩, ⟨33 , 22⟩)


The ordered 2-state communicating classes follow:

    (⟨ 0 , 1⟩, ⟨1 , 0⟩)


    (⟨ 0 , 2⟩, ⟨2 , 0⟩)


    (⟨ 1 , 11⟩, ⟨11 , 1⟩)


    (⟨ 2 , 22⟩, ⟨22 , 2⟩)


    (⟨ 11 , 31⟩, ⟨31 , 11⟩)


    (⟨ 22 , 32⟩, ⟨32 , 22⟩)


    (⟨ 31 , 33⟩, ⟨33 , 31⟩)


    (⟨ 32 , 33⟩, ⟨33 , 32⟩)


The ordered 1-state communicating classes follow:

    (⟨ 0 , 0⟩)


    (⟨ 11 , 11⟩)


    (⟨ 22 , 22⟩)


    (⟨ 33 , 33⟩)




 §.§ Presenting LQ


L_Q is the direct sum of one 9 × 9 block

    ℒ_9 =


    [
    [    *  D_1  D_2 D_10 D_11 D_10 D_11 D_21 D_21;  D_3    *    0 D_12    0 D_13    0    0    0;  D_4    0    *    0 D_14    0 D_14    0    0;  D_5  D_6    0    *    0 D_15 D_16 D_22    0;  D_7    0  D_8    0    * D_15 D_18    0 D_23;  D_5  D_6    0 D_15 D_16    *    0    0 D_22;  D_7    0  D_8 D_17 D_18    0    * D_23    0;  D_9    0    0 D_19    0    0 D_20    *    0;  D_9    0    0    0 D_20 D_19    0    0    *;      ]]


where 

    D_1 = q^4n+q^4-2q^2n+2/(q^12+4q^10+6q^8+4q^6+q^4)q^2n   D_2 = q^12+q^4n+8-2q^2n+10/(q^8+4q^6+6q^4+4q^2+1)q^2n)



    D_3 = q^4n-2q^2n+2+q^4/q^2n   D_4 =q^4n-2q^2n+2+q^4/q^2n+4



    D_5 = q^4n(2q^2-1)+2q^2n(q^6-q^4+q^2)+2q^6-q^8/(q^4+q^2)q^2n   D_6 = 2q^2n-q^2+1/(q^4+2q^2+1)q^2n



    D_7 = q^4n(2q^2-1)+2q^2n(q^6-q^4+q^2)+2q^6-q^8/(q^6+q^4)q^2n   D_8 = q^2n(q^4-q^2)+2q^4/q^4+2q^2+1



    D_9 = q^4n-2q^2n+2+q^4/q^2n+2   D_10 = 2q^2n(q^6-q^4+q^2)+q^4n(2q^2-1)+2q^6-q^8/(q^10+3q^8+3q^6+q^4)q^2n



    D_11 = q^4n(2q^4-q^2)+2q^2n(q^8-q^6+q^4)+2q^8-q^10/(q^6+3q^4+3q^2+1)q^2n   D_12 = 2q^2n+2-q^2-q^4/q^2n



    D_13 = 2q^2n+2-q^4+q^2/q^2n   D_14 = q^2n(q^2-1)+2q^2/q^4



    D_15 = q^4n-2q^2n+2+q^4/(q^6+2q^4+q^2)q^2n   D_16 = q^4n+4-2q^2n+6+q^8/(q^4+2q^2+1)q^2n



    D_17 = q^4n-2q^2n+2+q^4/(q^8+2q^6+q^4)q^2n   D_18 = q^4n+2-2q^2n+4+q^6/(q^4+2q^2+1)q^2n



    D_19 = 2q^2n+1-q^2/q^2n   D_20 = q^2n(q^2-1)+2q^2/q^2



    D_21 = q^6+q^4n+2-2q^2n+4/(q^8+4q^6+6q^4+4q^2+1)q^2n   D_22 = 2q^2n+6+q^6-q^8/(q^4+2q^2+1)q^2n



    D_23 = (q^2-1)q^2n+2q^2/q^8+2q^6+q^4



and ℒ_9 corresponds to the 9-state communicating class listed in Section <ref>,

four 6 × 6 blocks

    ℒ_6 =


    [
    [    *  D_1  D_2 D_14 D_15 D_16;  D_3    *  D_4 D_17 D_18 D_19;  D_5  D_6    *    0 D_20    0;  D_7  D_8    0    *    0 D_21;  D_9 D_10 D_11    0    *    0; D_12 D_13    0 D_22    0    *;      ]]


where 


    D_1 = 2q^6+(2q^2-1)q^4n+2q^2n(q^6-q^4+q^2)-q^8/(q^4+2q^2+1)q^2n   D_2 = q^4n+2q^2n+4+q^4+q^2-q^6/(q^8+3q^6+3q^4+q^2)q^2n



    D_3 = 2q^6+(2q^2-1)q^4n+2q^2n(q^6-q^4+q^2)-q^8/(q^8+2q^6+q^4)q^2n   D_4 = q^4+q^4n-2q^2n+2/(q^10+3q^8+3q^6+q^4)q^2n



    D_5 = q^4n+2q^2n+4+q^4+q^2-q^6/(q^2+1)q^2n   D_6 = q^6+q^4n+2-2q^2n+4/(q^2+1)



    D_7 = q^4+q^4n-2q^2n+2/(q^6+q^4)q^2n   D_8 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^6+q^4)q^2n



    D_9 =  q^4+q^4n-2q^2n+2/(q^4+q^2)q^2n   D_10 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^4+q^2)q^2n



    D_11 = 2q^2n-q^2+1/(q^2+1)q^2n   D_12 = q^4n+2q^2n+4/(q^4+q^2)q^2n



    D_13 = q^4n-2q^2n+2+q^4/(q^2+1)q^2n   D_14 = q^10+q^4n+6-2q^2n+8/(q^6+3q^4+3q^2+1)q^2n



    D_15 = q^4+q^4n-2q^2n+2/(q^6+3q^4+3q^2+1)q^2n   D_16 = q^8+q^6+q^4n+4+2q^2n+8-q^10/(q^6+3q^4+3q^2+1)q^2n



    D_17 = q^8+(q^6+q^4-q^2)q^4n+2q^2n+4/(q^6+3q^4+3q^2+1)q^2n   D_18 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^10+3q^8+3q^6+q^4)q^2n)



    D_19 = q^6+q^4n+2-2q^2n+4/(q^6+3q^4+3q^2+1)q^2n   D_20 = 2q^2n+4+q^4-q^6/(q^2+1)q^2n



    D_21 = (q^2-1)q^2n+2q^2/q^6+q^4   D_22 = (q^2-1)q^2n+2q^2/q^2+1


and ℒ_6 corresponds to the 6-state communicating classes listed in Section <ref>,

four 4 × 4 blocks

    ℒ_4 = 1/(q^4+2q^2+1)q^2nM


where 


    M =


    [
    [   * D_1 D_1 D_6; D_2   * D_3 D_7; D_2 D_3   * D_7; D_4 D_5 D_5   *;     ]]


where 


    D_1 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/q^4   D_2 = q^6+(q^4+q^2-1)q^4n+2q^2n+2



    D_3 = q^4+q^4n-2q^2n+2   D_4 = q^8+q^4n+4-2q^2n+6



    D_5 = q^4n+2+2q^2n+6+q^6+q^4-q^8    D_6 = q^4+q^4n-2q^2n+2/q^4



    D_7 = 2q^2n+4+q^4n+q^4+q^2-q^6/q^2


and ℒ_4 corresponds to the 4-state communicating classes listed in Section <ref>,

four 3 × 3 blocks

    ℒ_3 =  [                           * q^2+q^4n/(q^6+2q^4+q^2)q^2n q^6+q^4n+4/(q^4+2q^2+1)q^2n;               q^2+q^4n/q^2n                           *                           0;             q^2+q^4n/q^2n+2                           0                           * ]

corresponding to the 3-state communicating classes listed in Section <ref>,
eight 2 × 2 blocks

    ℒ_2 = [                      * q^4+q^4n+2/(q^2+1)q^2n; q^2+q^4n/(q^4+q^2)q^2n                      * ]

corresponding to the 2-state communicating classes listed in Section <ref>,

and four 1 × 1 blocks with entry 0 corresponding to the 1-state communicating classes listed in Section <ref>. Here, the diagonal entries are choosen so that the rows sum to 0. To summarize, the generator matrix is

    L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0]

with respect to the ordered basis that is the concatenation of all of the above communicating classes in the order they are presented.



 §.§ Ordering of States and Permutations of 
Lm and Lq




  §.§.§ Gamma State Ordering

The assumed ordering of states for four Γ lattice sites is Ω⊗Ω. To be more concise we represent the state (x_1,x_2,x_3,x_4) as x_1x_2x_3x_4.

Assumed Ordering:

(3030,   3021,   3003,   3012,   3010,   3001,   3020,   3002,   3031,   3013,   3032,   3023,   3000,   3011,   3022,   3033,   2130,   2121,   2103,   2112,   2110,   2101,   2120,   2102,   2131,   2113,   2132,   2123,   2100,   2111,   2122,   2133,   0330,   0321,   0303,   0312,   0310,   0301,   0320,   0302,   0331,   0313,   0332,   0323,   0300,   0311,   0322,   0333,   1230,   1221,   1203,   1212,   1210,   1201,   1220,   1202,   1231,   1213,   1232,   1223,   1200,   1211,   1222,   1233,   1030,   1021,   1003,   1012,   1010,   1001,   1020,   1002,   1031,   1013,   1032,   1023,   1000,   1011,   1022,   1033,   0130,   0121,   0103,   0112,   0110,   0101,   0120,   0102,   0131,   0113,   0132,   0123,   0100,   0111,   0122,   0133,   2030,   2021,   2003,   2012,   2010,   2001,   2020,   2002,   2031,   2013,   2032,   2023,   2000,   2011,   2022,   2033,   0230,   0221,   0203,   0212,   0210,   0201,   0220,   0202,   0231,   0213,   0232,   0223,   0200,   0211,   0222,   0233,   3130,   3121,   3103,   3112,   3110,   3101,   3120,   3102,   3131,   3113,   3132,   3123,   3100,   3111,   3122,   3133,   1330,   1321,   1303,   1312,   1310,   1301,   1320,   1302,   1331,   1313,   1332,   1323,   1300,   1311,   1322,   1333,   3230,   3221,   3203,   3212,   3210,   3201,   3220,   3202,   3231,   3213,   3232,   3223,   3200,   3211,   3222,   3233,   2330,   2321,   2303,   2312,   2310,   2301,   2320,   2302,   2331,   2313,   2332,   2323,   2300,   2311,   2322,   2333,   0030,   0021,   0003,   0012,   0010,   0001,   0020,   0002,   0031,   0013,   0032,   0023,   0000,   0011,   0022,   0033,   1130,   1121,   1103,   1112,   1110,   1101,   1120,   1102,   1131,   1113,   1132,   1123,   1100,   1111,   1122,   1133,   2230,   2221,   2203,   2212,   2210,   2201,   2220,   2202,   2231,   2213,   2232,   2223,   2200,   2211,   2222,   2233,   3330,   3321,   3303,   3312,   3310,   3301,   3320,   3302,   3331,   3313,   3332,   3323,   3300,   3311,   3322,   3333 )

Diagonal Ordering:

(3300,   3210,   3030,   3120,   3100,   3010,   3200,   3020,   3310,   3130,   3320,   3230,   3000,   3110,   3220,   3330,   2301,   2211,   2031,   2121,   2101,   2011,   2201,   2021,   2311,   2131,   2321,   2231,   2001,   2111,   2221,   2331,   0303,   0213,   0033,   0123,   0103,   0013,   0203,   0023,   0313,   0133,   0323,   0233,   0003,   0113,   0223,   0333,   1302,   1212,   1032,   1122,   1102,   1012,   1202,   1022,   1312,   1132,   1322,   1232,   1002,   1112,   1222,   1332,   1300,   1210,   1030,   1120,   1100,   1010,   1200,   1020,   1310,   1130,   1320,   1230,   1000,   1110,   1220,   1330,   0301,   0211,   0031,   0121,   0101,   0011,   0201,   0021,   0311,   0131,   0321,   0231,   0001,   0111,   0221,   0331,   2300,   2210,   2030,   2120,   2100,   2010,   2200,   2020,   2310,   2130,   2320,   2230,   2000,   2110,   2220,   2330,   0302,   0212,   0032,   0122,   0102,   0012,   0202,   0022,   0312,   0132,   0322,   0232,   0002,   0112,   0222,   0332,   3301,   3211,   3031,   3121,   3101,   3011,   3201,   3021,   3311,   3131,   3321,   3231,   3001,   3111,   3221,   3331,   1303,   1213,   1033,   1123,   1103,   1013,   1203,   1023,   1313,   1133,   1323,   1233,   1003,   1113,   1223,   1333,   3302,   3212,   3032,   3122,   3102,   3012,   3202,   3022,   3312,   3132,   3322,   3232,   3002,   3112,   3222,   3332,   2303,   2213,   2033,   2123,   2103,   2013,   2203,   2023,   2313,   2133,   2323,   2233,   2003,   2113,   2223,   2333,   0300,   0210,   0030,   0120,   0100,   0010,   0200,   0020,   0310,   0130,   0320,   0230,   0000,   0110,   0220,   0330,   1301,   1211,   1031,   1121,   1101,   1011,   1201,   1021,   1311,   1131,   1321,   1231,   1001,   1111,   1221,   1331,   2302,   2212,   2032,   2122,   2102,   2012,   2202,   2022,   2312,   2132,   2322,   2232,   2002,   2112,   2222,   2332,   3303,   3213,   3033,   3123,   3103,   3013,   3203,   3023,   3313,   3133,   3323,   3233,   3003,   3113,   3223,   3333 )




  §.§.§ gamma State Ordering

The assumed ordering of states for two γ lattice sites is χ⊗χ. To be more concise we represent the state (x_1,x_2) as x_1_x_2.

Assumed Ordering:

(0_0,   0_1,   0_2,   0_11,   0_3,   0_22,   0_31,   0_32,   0_33,   1_0,   1_1,   1_2,   1_11,   1_3,   1_22,   1_31,   1_32,   1_33,   2_0,   2_1,   2_2,   2_11,   2_3,   2_22,   2_31,   2_32,   2_33,   11_0,   11_1,   11_2,   11_11,   11_3,   11_22,   11_31,   11_32,   11_33,   3_0,   3_1,   3_2,   3_11,   3_3,   3_22,   3_31,   3_32,   3_33,   22_0,   22_1,   22_2,   22_11,   22_3,   22_22,   22_31,   22_32,   22_33,   31_0,   31_1,   31_2,   31_11,   31_3,   31_22,   31_31,   31_32,   31_33,   32_0,   32_1,   32_2,   32_11,   32_3,   32_22,   32_31,   32_32,   32_33,   33_0,   33_1,   33_2,   33_11,   33_3,   33_22,   33_31,   33_32,   33_33 )

Diagonal Ordering:

 (3_3,   0_33,   33_0,   1_32,   32_1,   2_31,   31_2,   11_22,   22_11,   1_3,   3_1,   0_31,   31_0,   2_11,   11_2,   2_3,   3_2,   0_32,   32_0,   1_22,   22_1,   3_31,   31_3,   1_33,   33_1,   11_32,   32_11,   3_32,   32_3,   2_33,   33_2,   22_31,   31_22,   3_0,   1_2,   2_1,   0_3,   31_1,   11_3,   3_11,   1_31,   32_2,   22_3,   3_22,   2_32,   33_3,   31_32,   32_31,   3_33,   1_1,   0_11,   11_0,   2_2,   0_22,   22_0,   31_31,   11_33,   33_11,   32_32,   22_33,   33_22,   0_1,   1_0,   0_2,   2_0,   1_11,   11_1,   2_22,   22_2,   11_31,   31_11,   22_32,   32_22,   31_33,   33_31,   32_33,   33_32,   0_0,   11_11,   22_22,   33_33 )




 §.§ Limits of Components of 
Lp2
 

Recall that L^(2)_p is given in section 2.2 “Probabilistic definitions” of <cit.> as the direct sum of three distinct block matrices L_1, L_2, and four 1 × 1 zero matrices.

We list the limits for q^-2n L_1 and q^-2n L_2 for q > 1. The limits follow similarly when multiplying  L_1 and L_2 by q^2n for 0 < q < 1.

Take q > 1.


    lim_n →∞ (q^-2n L_1) = 
    [ -2q^2-1/q^4   q^2-1/q^4       1/q^4   q^2-1/q^4; q^2 - 1/q^2          -1           0       1/q^2;           1           0          -1           0; q^2 - 1/q^2       1/q^2           0          -1;             ]


    lim_n →∞ (q^-2n L_2) = 
    [ -1/q^2  1/q^2;      1     -1;        ]




 §.§ Diagonalization for Communicating Classes


We show that for each communicating class ℒ_i where i ∈{3,4,6,9} the matrix of right eigenvectors is invertible for q > 0 and q ≠ 1 , and therefore that all the eigenvectors are linearly independent which implies diagonalizability.

Consider  𝒫_9,

    𝒫_9 =


    [                                     1             - q^4/q^8 + q^6 + q^2 + 1             - q^4/q^8 + q^6 + q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1     - q^4 + 2 q^2 - 1/q^4 + 2 q^2 + 1;                                     1                           q^6/q^6 + 1                           q^6/q^6 + 1                           q^4/q^4 - 1                           q^4/q^4 - 1                                   q^6                         - q^6/q^2 - 1                         - q^6/q^2 - 1                                 - q^4;                                     1                             1/q^6 + 1                             1/q^6 + 1                           - 1/q^4 - 1                           - 1/q^4 - 1                                 1/q^6                           1/q^6 - q^4                           1/q^6 - q^4                               - 1/q^4;                                     1 q^10 - q^6 + q^4/q^10 - q^6 + q^4 - 1            - q^6/q^10 - q^6 + q^4 - 1   q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1               q^4/q^6 + q^4 - q^2 - 1                         - q^4/q^2 + 1 - q^8 + q^6 + q^4/q^6 + q^4 - q^2 - 1               q^4/q^6 + q^4 - q^2 - 1                   - q^4 + q^2/q^2 + 1;                                     1              q^4/q^10 - q^6 + q^4 - 1  - q^6 + q^4 - 1/q^10 - q^6 + q^4 - 1             - q^2/q^6 + q^4 - q^2 - 1     q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1                         - 1/q^4 + q^2             - q^2/q^6 + q^4 - q^2 - 1 - q^4 - q^2 + 1/q^8 + q^6 - q^4 - q^2                     q^2 - 1/q^4 + q^2;                                     1            - q^6/q^10 - q^6 + q^4 - 1 q^10 - q^6 + q^4/q^10 - q^6 + q^4 - 1               q^4/q^6 + q^4 - q^2 - 1   q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1                         - q^4/q^2 + 1               q^4/q^6 + q^4 - q^2 - 1 - q^8 + q^6 + q^4/q^6 + q^4 - q^2 - 1                   - q^4 + q^2/q^2 + 1;                                     1  - q^6 + q^4 - 1/q^10 - q^6 + q^4 - 1              q^4/q^10 - q^6 + q^4 - 1     q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1             - q^2/q^6 + q^4 - q^2 - 1                         - 1/q^4 + q^2 - q^4 - q^2 + 1/q^8 + q^6 - q^4 - q^2             - q^2/q^6 + q^4 - q^2 - 1                     q^2 - 1/q^4 + q^2;                                     1                                     1                                     0                                     1                                     0                                     1                                     1                                     0                                     1;                                     1                                     0                                     1                                     0                                     1                                     1                                     0                                     1                                     1 ]


It follows that 

    (𝒫_9) = (q^4 + 1)^5(q^2 - q + 1)^5(q^2 + q + 1)^5/q^6(q - 1)^3(q + 1)^3(q^2 + 1)^9(q^4 - q^2 + 1)
 
For q > 0 and q ≠ 1, the (𝒫_9) is not 0 and is defined. Therefore, 𝒫_9 is invertible for q > 0 and q ≠ 1.

Consider  𝒫_6,

    𝒫_6 = [                                   1                       - q^4/q^2 + 1             q^4/q^6 + q^4 - q^2 - 1 q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1                       - q^4/q^2 + 1                 - q^4 + q^2/q^2 + 1;                                   1                       - q^4/q^2 + 1   q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1           - q^2/q^6 + q^4 - q^2 - 1                           1/q^2 + 1                   q^2 - 1/q^4 + q^2;                                   1                                 q^6                         q^4/q^4 - 1                         q^4/q^4 - 1                                 q^6                               - q^4;                                   1                                   1                         - 1/q^4 - 1                         - 1/q^4 - 1                             - 1/q^4                             - 1/q^4;                                   1                                 q^6                                   1                                   0                               - q^2                                   1;                                   1                                   1                                   0                                   1                                   1                                   1 ]


It follows that 

    (𝒫_6) = -(q^4+1)^4(q^2-q+1)^2(q^2+q+1)^2/q^4(q-1)(q+1)(q^2+1)^3
 
For q > 0 and q ≠ 1, the (𝒫_6) is not 0 and is defined. Therefore, 𝒫_6 is invertible for q > 0 and q ≠ 1.

Consider 𝒫_4,

    𝒫_4 = [         1         0    -1/q^4     1/q^8;         1        -1 q^4-1/q^4    -1/q^4;         1         1         0    -1/q^4;         1         0         1         1 ]

It follows that 

    (𝒫_4) = - q^12+3q^8+3q^4+1/q^12
 
For q > 0, the (𝒫_4) is not 0. Therefore, 𝒫_4 is invertible for q > 0.

Consider 𝒫_3,

    𝒫_3 = [             1    -q^4/q^2+1 q^2-q^4/q^2+1;             1           q^6          -q^4;             1             1             1 ]

It follows that 

    (𝒫_3) = q^10+q^8+2q^6+q^4+q^2/q^2+1
 
For q > 0, the (𝒫_3) is not 0. Therefore, 𝒫_3 is invertible for q > 0.



 §.§ Ordering of Basis Vectors to Block pi(W tensor W)(C)
 
Denote v_i to be the i^th basis vector of W. Then, the blocked matrix π_W ⊗ W(C) is with respect to ordered basis: 

       v_1 ⊗v_1,
v_1 ⊗v_20,
v_2 ⊗v_19,
v_4 ⊗v_17,
v_5 ⊗v_16,
v_8 ⊗v_13,
v_9 ⊗v_9,
v_3 ⊗v_18,
v_6 ⊗v_15, 
 
   v_7 ⊗v_14,
v_9 ⊗v_11,
v_11 ⊗v_9,
v_11 ⊗v_11,
v_13 ⊗v_8,
v_10 ⊗v_12,
v_14 ⊗v_7,
v_16 ⊗v_5,
v_12 ⊗v_10, 
 
   v_15 ⊗v_6,
v_17 ⊗v_4,
v_18 ⊗v_3,
v_19 ⊗v_2,
v_20 ⊗v_1,
v_1 ⊗v_13,
v_2 ⊗v_9,
v_2 ⊗v_11,
v_4 ⊗v_7, 
 
   v_5 ⊗v_6,
v_8 ⊗v_3,
v_9 ⊗v_2,
v_3 ⊗v_8,
v_6 ⊗v_5,
v_7 ⊗v_4,
v_11 ⊗v_2,
v_13 ⊗v_1,
v_1 ⊗v_16, 
 
   v_2 ⊗v_14,
v_4 ⊗v_9,
v_4 ⊗v_11,
v_5 ⊗v_10,
v_8 ⊗v_6,
v_9 ⊗v_4,
v_6 ⊗v_8,
v_11 ⊗v_4,
v_10 ⊗v_5, 
 
   v_14 ⊗v_2,
v_16 ⊗v_1,
v_1 ⊗v_17,
v_2 ⊗v_15,
v_4 ⊗v_12,
v_5 ⊗v_9,
v_5 ⊗v_11,
v_8 ⊗v_7,
v_9 ⊗v_5, 
 
   v_7 ⊗v_8,
v_11 ⊗v_5,
v_12 ⊗v_4,
v_15 ⊗v_2,
v_17 ⊗v_1,
v_1 ⊗v_19,
v_2 ⊗v_18,
v_4 ⊗v_15,
v_5 ⊗v_14, 
 
   v_8 ⊗v_9,
v_8 ⊗v_11,
v_9 ⊗v_8,
v_11 ⊗v_8,
v_14 ⊗v_5,
v_15 ⊗v_4,
v_18 ⊗v_2,
v_19 ⊗v_1,
v_2 ⊗v_16, 
 
   v_3 ⊗v_14,
v_4 ⊗v_13,
v_6 ⊗v_9,
v_6 ⊗v_11,
v_7 ⊗v_10,
v_9 ⊗v_6,
v_11 ⊗v_6,
v_13 ⊗v_4,
v_14 ⊗v_3, 
 
   v_16 ⊗v_2,
v_10 ⊗v_7,
v_2 ⊗v_17,
v_3 ⊗v_15,
v_5 ⊗v_13,
v_6 ⊗v_12,
v_7 ⊗v_9,
v_7 ⊗v_11,
v_9 ⊗v_7, 
 
   v_11 ⊗v_7,
v_13 ⊗v_5,
v_12 ⊗v_6,
v_15 ⊗v_3,
v_17 ⊗v_2,
v_2 ⊗v_20,
v_3 ⊗v_19,
v_6 ⊗v_17,
v_7 ⊗v_16, 
 
   v_9 ⊗v_13,
v_11 ⊗v_13,
v_13 ⊗v_9,
v_13 ⊗v_11,
v_16 ⊗v_7,
v_17 ⊗v_6,
v_19 ⊗v_3,
v_20 ⊗v_2,
v_4 ⊗v_19, 
 
   v_6 ⊗v_18,
v_8 ⊗v_16,
v_9 ⊗v_14,
v_10 ⊗v_15,
v_11 ⊗v_14,
v_14 ⊗v_9,
v_14 ⊗v_11,
v_16 ⊗v_8,
v_18 ⊗v_6, 
 
   v_19 ⊗v_4,
v_15 ⊗v_10,
v_4 ⊗v_20,
v_6 ⊗v_19,
v_9 ⊗v_16,
v_10 ⊗v_17,
v_11 ⊗v_16,
v_14 ⊗v_13,
v_16 ⊗v_9, 
 
   v_13 ⊗v_14,
v_16 ⊗v_11,
v_17 ⊗v_10,
v_19 ⊗v_6,
v_20 ⊗v_4,
v_5 ⊗v_19,
v_7 ⊗v_18,
v_8 ⊗v_17,
v_9 ⊗v_15, 
 
   v_11 ⊗v_15,
v_12 ⊗v_14,
v_15 ⊗v_9,
v_15 ⊗v_11,
v_17 ⊗v_8,
v_14 ⊗v_12,
v_18 ⊗v_7,
v_19 ⊗v_5,
v_5 ⊗v_20, 
 
   v_7 ⊗v_19,
v_9 ⊗v_17,
v_11 ⊗v_17,
v_12 ⊗v_16,
v_15 ⊗v_13,
v_17 ⊗v_9,
v_13 ⊗v_15,
v_17 ⊗v_11,
v_16 ⊗v_12, 
 
   v_19 ⊗v_7,
v_20 ⊗v_5,
v_8 ⊗v_20,
v_9 ⊗v_19,
v_11 ⊗v_19,
v_14 ⊗v_17,
v_15 ⊗v_16,
v_18 ⊗v_13,
v_19 ⊗v_9, 
 
   v_13 ⊗v_18,
v_16 ⊗v_15,
v_17 ⊗v_14,
v_19 ⊗v_11,
v_20 ⊗v_8,
v_1 ⊗v_9,
v_2 ⊗v_8,
v_4 ⊗v_5,
v_5 ⊗v_4, 
 
   v_8 ⊗v_2,
v_9 ⊗v_1,
v_1 ⊗v_11,
v_11 ⊗v_1,
v_2 ⊗v_13,
v_3 ⊗v_9,
v_3 ⊗v_11,
v_6 ⊗v_7,
v_7 ⊗v_6, 
 
   v_9 ⊗v_3,
v_11 ⊗v_3,
v_13 ⊗v_2,
v_4 ⊗v_16,
v_6 ⊗v_14,
v_9 ⊗v_10,
v_10 ⊗v_9,
v_10 ⊗v_11,
v_11 ⊗v_10, 
 
   v_14 ⊗v_6,
v_16 ⊗v_4,
v_5 ⊗v_17,
v_7 ⊗v_15,
v_9 ⊗v_12,
v_11 ⊗v_12,
v_12 ⊗v_9,
v_12 ⊗v_11,
v_15 ⊗v_7, 
 
   v_17 ⊗v_5,
v_8 ⊗v_19,
v_9 ⊗v_18,
v_11 ⊗v_18,
v_14 ⊗v_15,
v_15 ⊗v_14,
v_18 ⊗v_9,
v_18 ⊗v_11,
v_19 ⊗v_8, 
 
   v_9 ⊗v_20,
v_13 ⊗v_19,
v_16 ⊗v_17,
v_17 ⊗v_16,
v_19 ⊗v_13,
v_20 ⊗v_9,
v_11 ⊗v_20,
v_20 ⊗v_11,
v_1 ⊗v_6, 
 
   v_2 ⊗v_4,
v_4 ⊗v_2,
v_6 ⊗v_1,
v_1 ⊗v_7,
v_2 ⊗v_5,
v_5 ⊗v_2,
v_7 ⊗v_1,
v_1 ⊗v_14,
v_4 ⊗v_8, 
 
   v_8 ⊗v_4,
v_14 ⊗v_1,
v_1 ⊗v_15,
v_5 ⊗v_8,
v_8 ⊗v_5,
v_15 ⊗v_1,
v_2 ⊗v_6,
v_3 ⊗v_4,
v_4 ⊗v_3, 
 
   v_6 ⊗v_2,
v_2 ⊗v_7,
v_3 ⊗v_5,
v_5 ⊗v_3,
v_7 ⊗v_2,
v_2 ⊗v_10,
v_4 ⊗v_6,
v_6 ⊗v_4,
v_10 ⊗v_2, 
 
   v_2 ⊗v_12,
v_5 ⊗v_7,
v_7 ⊗v_5,
v_12 ⊗v_2,
v_3 ⊗v_16,
v_6 ⊗v_13,
v_13 ⊗v_6,
v_16 ⊗v_3,
v_3 ⊗v_17, 
 
   v_7 ⊗v_13,
v_13 ⊗v_7,
v_17 ⊗v_3,
v_4 ⊗v_14,
v_8 ⊗v_10,
v_10 ⊗v_8,
v_14 ⊗v_4,
v_4 ⊗v_18,
v_8 ⊗v_14, 
 
   v_14 ⊗v_8,
v_18 ⊗v_4,
v_5 ⊗v_15,
v_8 ⊗v_12,
v_12 ⊗v_8,
v_15 ⊗v_5,
v_5 ⊗v_18,
v_8 ⊗v_15,
v_15 ⊗v_8, 
 
   v_18 ⊗v_5,
v_6 ⊗v_16,
v_10 ⊗v_13,
v_13 ⊗v_10,
v_16 ⊗v_6,
v_6 ⊗v_20,
v_13 ⊗v_16,
v_16 ⊗v_13,
v_20 ⊗v_6, 
 
   v_7 ⊗v_17,
v_12 ⊗v_13,
v_13 ⊗v_12,
v_17 ⊗v_7,
v_7 ⊗v_20,
v_13 ⊗v_17,
v_17 ⊗v_13,
v_20 ⊗v_7,
v_10 ⊗v_19, 
 
   v_14 ⊗v_16,
v_16 ⊗v_14,
v_19 ⊗v_10,
v_12 ⊗v_19,
v_15 ⊗v_17,
v_17 ⊗v_15,
v_19 ⊗v_12,
v_14 ⊗v_19,
v_16 ⊗v_18, 
 
   v_18 ⊗v_16,
v_19 ⊗v_14,
v_14 ⊗v_20,
v_16 ⊗v_19,
v_19 ⊗v_16,
v_20 ⊗v_14,
v_15 ⊗v_19,
v_17 ⊗v_18,
v_18 ⊗v_17, 
 
   v_19 ⊗v_15,
v_15 ⊗v_20,
v_17 ⊗v_19,
v_19 ⊗v_17,
v_20 ⊗v_15,
v_1 ⊗v_3,
v_2 ⊗v_2,
v_3 ⊗v_1,
v_1 ⊗v_10, 
 
   v_4 ⊗v_4,
v_10 ⊗v_1,
v_1 ⊗v_12,
v_5 ⊗v_5,
v_12 ⊗v_1,
v_1 ⊗v_18,
v_8 ⊗v_8,
v_18 ⊗v_1,
v_3 ⊗v_10, 
 
   v_6 ⊗v_6,
v_10 ⊗v_3,
v_3 ⊗v_12,
v_7 ⊗v_7,
v_12 ⊗v_3,
v_3 ⊗v_20,
v_13 ⊗v_13,
v_20 ⊗v_3,
v_10 ⊗v_18, 
 
   v_14 ⊗v_14,
v_18 ⊗v_10,
v_10 ⊗v_20,
v_16 ⊗v_16,
v_20 ⊗v_10,
v_12 ⊗v_18,
v_15 ⊗v_15,
v_18 ⊗v_12,
v_12 ⊗v_20, 
 
   v_17 ⊗v_17,
v_20 ⊗v_12,
v_18 ⊗v_20,
v_19 ⊗v_19,
v_20 ⊗v_18,
v_1 ⊗v_2,
v_2 ⊗v_1,
v_1 ⊗v_4,
v_4 ⊗v_1, 
 
   v_1 ⊗v_5,
v_5 ⊗v_1,
v_1 ⊗v_8,
v_8 ⊗v_1,
v_2 ⊗v_3,
v_3 ⊗v_2,
v_3 ⊗v_6,
v_6 ⊗v_3,
v_3 ⊗v_7, 
 
   v_7 ⊗v_3,
v_3 ⊗v_13,
v_13 ⊗v_3,
v_4 ⊗v_10,
v_10 ⊗v_4,
v_5 ⊗v_12,
v_12 ⊗v_5,
v_6 ⊗v_10,
v_10 ⊗v_6, 
 
   v_7 ⊗v_12,
v_12 ⊗v_7,
v_8 ⊗v_18,
v_18 ⊗v_8,
v_10 ⊗v_14,
v_14 ⊗v_10,
v_10 ⊗v_16,
v_16 ⊗v_10,
v_12 ⊗v_15, 
 
   v_15 ⊗v_12,
v_12 ⊗v_17,
v_17 ⊗v_12,
v_13 ⊗v_20,
v_20 ⊗v_13,
v_14 ⊗v_18,
v_18 ⊗v_14,
v_15 ⊗v_18,
v_18 ⊗v_15, 
 
   v_16 ⊗v_20,
v_20 ⊗v_16,
v_17 ⊗v_20,
v_20 ⊗v_17,
v_18 ⊗v_19,
v_19 ⊗v_18,
v_19 ⊗v_20,
v_20 ⊗v_19,
v_3 ⊗v_3, 
 
   v_10 ⊗v_10,
v_12 ⊗v_12,
v_18 ⊗v_18,
v_20 ⊗v_20





 §.§ Algebraically-produced Markov Generator
 
This generator is produced as described in Section <ref>, with the additional step of deleting the ith row and column of the matrix a^-1G^-1π_W ⊗ W(C)G - Id if g_i = 0, or if row i or column i of the matrix contains a negative off-diagonal entry, as those lack a probabilistic interpretation.

The authors note that the polynomials below may be expressible as q-hypergeometric series, but we do not investigate this possibility in this paper. Recall that a = q^12 + q^2 + q^-2 + q^-12, and let the matrix entry * denote, as in Section <ref>, -1 times the sum of the other elements in its row. We may express the Markov generator produced algebraically as the direct sum of one 9 × 9 block:

    a^-2[    *  B_1  B_1  B_2    0    0    0    0    0;  B_3    *  B_4  B_5  B_6  B_7    0    0    0;  B_3  B_4    *  B_5    0    0  B_6  B_7    0;  B_8  B_9  B_9    * B_10 B_11 B_10 B_11 B_12;    0 B_13    0 B_14    * B_15    0    0    0;    0 B_16    0 B_17 B_18    *    0 B_19 B_20;    0    0 B_13 B_14    0    0    * B_15    0;    0    0 B_16 B_17    0 B_19 B_18    * B_20;    0    0    0 B_21    0 B_22    0 B_22    * ]

where

    B_1    =
    q^22a^2-q^20a^2-2q^18a^2+2q^16a^2+3q^14a^2-q^12a^2-4q^10a^2+2q^6a^2
    
        B_2    =
    q^20a^2-2q^16a^2-2q^14a^2+q^12a^2+4q^10a^2+q^8a^2-2q^6a^2-2q^4a^2+1a^2
    
        B_3    =
    q^24a^2-3q^22a^2+3q^20a^2-q^18a^2+2q^16a^2-4q^14a^2+2q^12a^2
    
        B_4    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_5    =
    a^2 ·2q^16-3q^14-q^12+2q^10+2q^8-q^6-3q^4+2q^2/q^20-q^16+q^12+q^10+q^8-q^4+1
    
        B_6    =
    q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+2q^10a^2-4q^8a^2+2q^6a^2
    
        B_7    =
    q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+4q^8a^2-2q^6a^2+q^4a^2-2q^2a^2+1a^2
    
        B_8    =
    q^24a^2-4q^22a^2+6q^20a^2-4q^18a^2+q^16a^2
    
        B_9    =
    2q^22a^2-5q^20a^2+3q^18a^2+q^16a^2+q^14a^2-4q^12a^2+2q^10a^2
    
        B_10   =
    q^18a^2-2q^16a^2-q^14a^2+4q^12a^2-q^10a^2-2q^8a^2+q^6a^2
    
        B_11   =
    2q^14a^2-4q^12a^2+q^10a^2+q^8a^2+3q^6a^2-5q^4a^2+2q^2a^2
    
        B_12   =
    q^8a^2-4q^6a^2+6q^4a^2-4q^2a^2+1a^2
    
        B_13   =
    q^24a^2-q^22a^2-2q^20a^2+2q^18a^2+3q^16a^2-q^14a^2-4q^12a^2+2q^8a^2
    
        B_14   =
    q^22a^2-2q^18a^2-2q^16a^2+q^14a^2+4q^12a^2+q^10a^2-2q^8a^2-2q^6a^2+q^2a^2
    
        B_15   =
    2q^16a^2-4q^12a^2-q^10a^2+3q^8a^2+2q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_16   =
    q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_17   =
    a^2 ·2q^18-3q^16-q^14+2q^12+2q^10-q^8-3q^6+2q^4/q^20-q^16+q^12+q^10+q^8-q^4+1
    
        B_18   =
    2q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+q^6a^2
    
        B_19   =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_20   =
    2q^12a^2-4q^10a^2+2q^8a^2-q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_21   =
    q^24a^2-2q^20a^2-2q^18a^2+q^16a^2+4q^14a^2+q^12a^2-2q^10a^2-2q^8a^2+q^4a^2
    
        B_22   =
    2q^18a^2-4q^14a^2-q^12a^2+3q^10a^2+2q^8a^2-2q^6a^2-q^4a^2+q^2a^2

a 6 × 6 block:

    a^-2[    *  B_1  B_2  B_3    0  B_4;  B_1    *  B_2    0  B_3  B_4;  B_5  B_5    *    0    0  B_6;  B_7    0    0    *  B_1  B_8;    0  B_7    0  B_1    *  B_8;  B_9  B_9 B_10 B_11 B_11    * ]

where

    B_1    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_2    =
    2q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-3q^4a^2+q^2a^2
    
        B_3    =
    q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_4    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    =
    2q^18a^2-2q^16a^2-2q^14a^2+q^12a^2+2q^10a^2-2q^6a^2+q^4a^2
    
        B_6    =
    q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_7    =
    q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_8    =
    q^18a^2-2q^16a^2+3q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_9    =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_10   =
    -q^18a^2+3q^16a^2-3q^14a^2+q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_11   =
    q^20a^2-2q^18a^2+2q^16a^2-3q^14a^2+3q^12a^2-q^10a^2+q^8a^2-2q^6a^2+q^4a^2

two 6 × 6 blocks:

    a^-2[    *  B_1  B_2  B_3    0    0;  B_4    *  B_5  B_6    0    0;  B_7  B_8    *  B_9    0  B_3; B_10 B_11 B_12    * B_13 B_14;    0    0    0 B_15    * B_16;    0    0 B_17 B_18 B_19    * ]

where

    B_1    =
    q^20a^2-3q^18a^2+3q^16a^2-q^14a^2+2q^12a^2-4q^10a^2+2q^8a^2
    
        B_2    =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_4    =
    q^24a^2-2q^22a^2+2q^18a^2+q^16a^2-2q^14a^2-2q^12a^2+2q^10a^2
    
        B_5    =
    2q^16a^2-2q^14a^2-2q^12a^2+q^10a^2+2q^8a^2-2q^4a^2+q^2a^2
    
        B_6    =
    q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_7    =
    q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_8    =
    2q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+q^6a^2
    
        B_9    =
    q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_10   =
    q^24a^2-3q^22a^2+3q^20a^2-2q^18a^2+3q^16a^2-3q^14a^2+q^12a^2
    
        B_11   =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_12   =
    -q^18a^2+3q^16a^2-3q^14a^2+q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_13   =
    q^18a^2-2q^16a^2+q^12a^2+q^10a^2-2q^6a^2+q^4a^2
    
        B_14   =
    q^14a^2-q^12a^2-q^10a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_15   =
    q^20a^2-q^18a^2-q^16a^2-q^14a^2+2q^12a^2+2q^10a^2-q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_16   =
    2q^14a^2-2q^12a^2-2q^10a^2+q^8a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_17   =
    q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_18   =
    q^20a^2-2q^18a^2+3q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-2q^6a^2-q^4a^2+q^2a^2
    
        B_19   =
    2q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2

a 6 × 6 block:

    a^-2[   * B_1 B_1   0   0   0; B_2   * B_3 B_4   0   0; B_2 B_5   *   0 B_4   0;   0 B_6   0   * B_7 B_8;   0   0 B_6 B_7   * B_8;   0   0   0 B_9 B_9   * ]

where

    B_1    =
    q^22a^2-2q^20a^2+2q^16a^2+q^14a^2-2q^12a^2-2q^10a^2+2q^8a^2
    
        B_2    =
    q^24a^2-3q^22a^2+3q^20a^2-q^18a^2+2q^16a^2-4q^14a^2+2q^12a^2
    
        B_3    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_4    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_6    =
    q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_7    =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_8    =
    2q^12a^2-4q^10a^2+2q^8a^2-q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_9    =
    2q^16a^2-2q^14a^2-2q^12a^2+q^10a^2+2q^8a^2-2q^4a^2+q^2a^2

four 6 × 6 blocks:

    a^-2[    *  B_1  B_2  B_3    0    0;  B_4    *  B_5  B_6    0    0;  B_7  B_8    *  B_9 B_10 B_11; B_12 B_13 B_14    * B_15 B_16;    0    0 B_17 B_18    * B_19;    0    0 B_20 B_21 B_22    * ]

where

    B_1    =
    q^20a^2-2q^18a^2+2q^14a^2+q^12a^2-2q^10a^2-2q^8a^2+2q^6a^2
    
        B_2    =
    q^22a^2-3q^18a^2+q^16a^2+2q^14a^2+q^12a^2-2q^10a^2-q^8a^2+q^6a^2-q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-q^16a^2-q^14a^2-q^12a^2+2q^10a^2+2q^8a^2-q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_4    =
    q^24a^2-2q^22a^2+2q^18a^2+q^16a^2-2q^14a^2-2q^12a^2+2q^10a^2
    
        B_5    =
    q^22a^2-q^20a^2-q^18a^2-q^16a^2+2q^14a^2+2q^12a^2-q^10a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_6    =
    q^20a^2-q^18a^2+q^16a^2-q^14a^2-2q^12a^2+q^10a^2+2q^8a^2+q^6a^2-3q^4a^2+1a^2
    
        B_7    =
    q^24a^2-2q^22a^2+2q^18a^2-q^14a^2-q^12a^2+q^10a^2
    
        B_8    =
    q^20a^2-2q^18a^2+q^14a^2+q^12a^2-2q^8a^2+q^6a^2
    
        B_9    =
    q^18a^2-q^16a^2-2q^12a^2+3q^10a^2-q^8a^2+2q^6a^2-4q^4a^2+2q^2a^2
    
        B_10   =
    q^18a^2-2q^16a^2+q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+2q^6a^2-2q^4a^2+q^2a^2
    
        B_11   =
    q^12a^2-3q^10a^2+3q^8a^2-2q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_12   =
    q^24a^2-3q^22a^2+3q^20a^2-2q^18a^2+3q^16a^2-3q^14a^2+q^12a^2
    
        B_13   =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_14   =
    2q^22a^2-4q^20a^2+2q^18a^2-q^16a^2+3q^14a^2-2q^12a^2-q^8a^2+q^6a^2
    
        B_15   =
    q^18a^2-2q^16a^2+q^12a^2+q^10a^2-2q^6a^2+q^4a^2
    
        B_16   =
    q^14a^2-q^12a^2-q^10a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_17   =
    q^24a^2-3q^20a^2+q^18a^2+2q^16a^2+q^14a^2-2q^12a^2-q^10a^2+q^8a^2-q^6a^2+q^4a^2
    
        B_18   =
    q^20a^2-q^18a^2-q^16a^2-q^14a^2+2q^12a^2+2q^10a^2-q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_19   =
    2q^14a^2-2q^12a^2-2q^10a^2+q^8a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_20   =
    q^24a^2-q^22a^2-q^20a^2-q^18a^2+2q^16a^2+2q^14a^2-q^12a^2-q^10a^2-q^8a^2+q^6a^2
    
        B_21   =
    q^22a^2-q^20a^2+q^18a^2-q^16a^2-2q^14a^2+q^12a^2+2q^10a^2+q^8a^2-3q^6a^2+q^2a^2
    
        B_22   =
    2q^18a^2-2q^16a^2-2q^14a^2+q^12a^2+2q^10a^2-2q^6a^2+q^4a^2

a 4 × 4 block:

    a^-2[   * B_1 B_2 B_3; B_1   * B_2 B_3; B_4 B_4   * B_3; B_5 B_5 B_6   * ]

where

    B_1    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_2    =
    2q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-3q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_4    =
    2q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_5    =
    q^22a^2-q^20a^2-q^16a^2+2q^12a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_6    =
    -q^18a^2+2q^16a^2-2q^12a^2+2q^8a^2-2q^4a^2+q^2a^2

two 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   * B_4 B_5; B_3 B_4   * B_5; B_6 B_7 B_7   * ]

where

    B_1    =
    q^20a^2-q^18a^2-q^16a^2+2q^12a^2-q^8a^2-q^4a^2+q^2a^2
    
        B_2    =
    q^14a^2-2q^12a^2+q^8a^2+q^6a^2-2q^2a^2+1a^2
    
        B_3    =
    q^24a^2-q^22a^2-2q^20a^2+3q^18a^2-q^16a^2+2q^14a^2-4q^12a^2+3q^10a^2-2q^8a^2+q^6a^2
    
        B_4    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_5    =
    q^18a^2-2q^16a^2+3q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_6    =
    q^24a^2-2q^22a^2+q^18a^2+q^16a^2-2q^12a^2+q^10a^2
    
        B_7    =
    q^22a^2-q^20a^2-q^16a^2+2q^12a^2-q^8a^2-q^6a^2+q^4a^2

two 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   * B_4 B_5; B_3 B_4   * B_5; B_6 B_7 B_7   * ]

where

    B_1    = q^20a^2-q^18a^2-2q^16a^2+3q^14a^2-q^12a^2+2q^10a^2-4q^8a^2+3q^6a^2-2q^4a^2+q^2a^2
    
        B_2    = q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+4q^8a^2-2q^6a^2+q^4a^2-2q^2a^2+1a^2
    
        B_3    = q^24a^2-q^22a^2-2q^20a^2+2q^18a^2+2q^16a^2-q^14a^2-2q^12a^2+q^8a^2
    
        B_4    = q^20a^2-q^18a^2-2q^16a^2+q^14a^2+2q^12a^2+q^10a^2-2q^8a^2-q^6a^2+q^4a^2
    
        B_5    = q^16a^2-2q^12a^2-q^10a^2+2q^8a^2+2q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_6    = q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_7    = q^22a^2-2q^20a^2+3q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-2q^8a^2-q^6a^2+q^4a^2

eight 4 × 4 blocks:

    a^-2[   * B_1 B_2   0; B_3   * B_4 B_5; B_3 B_6   * B_5;   0 B_7 B_1   * ]

where

    B_1    = q^22a^2-q^20a^2-q^18a^2+q^16a^2+q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_2    = q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_3    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_4    = q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_5    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_6    = q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_7    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2

four 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   *   0 B_4; B_3   0   * B_4; B_5 B_6 B_6   * ]

where

    B_1    = q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_2    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_3    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_4    = q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_5    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_6    = q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2

eight 3 × 3 blocks:

    a^-2[  B_1 -B_1    0;  B_2  B_3  B_4;    0  B_5 -B_5 ]

where

    B_1    = -q^22a^2+2q^18a^2-q^14a^2-q^8a^2+2q^4a^2-1a^2
    
        B_2    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_3    = -q^24a^2+2q^22a^2-q^20a^2-q^18a^2+2q^16a^2-q^14a^2-q^10a^2+2q^8a^2-q^6a^2-q^4a^2+2q^2a^2-1a^2
    
        B_4    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    = q^24a^2-2q^20a^2+q^16a^2+q^10a^2-2q^6a^2+q^2a^2

sixteen 2 × 2 blocks:

    a^-2[  B_1 -B_1;  B_2 -B_2 ]

where

    B_1    = -q^20a^2+q^18a^2+q^16a^2-q^14a^2-q^6a^2+q^4a^2+q^2a^2-1a^2
    
        B_2    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2

and five 1 × 1 blocks with entry 0.


Note that multiplying the 3 × 3 and 2 × 2 blocks of the algebraic generator by the scalar

    a^2/q^12·q^2-1+q^-2/(q-q^-1)^2 (q+q^-1) (q^7+q^-7)

yields the 3 × 3 and 2 × 2 blocks of L_Q (as described in Section <ref>) when n=2. Note additionally that no larger block of L_Q (with n=2) is a constant multiple of a block of the algebraic generator.





 §.§ Type A Ground State Transformation
 
The following notes were provided to us by Dr. Jeffrey Kuan to aid in our calculation of the ground state transformation.

Suppose C is a central element and H= (π_V ⊗π_V) (Δ (C)). Let v ∈ V be a highest-weight vector of a highest-weight representation V. Then,

    H(v ⊗ v) = av⊗ v

for some constant a. By the definition of a central element,

    Δ(F_j^k) C = C Δ(F_j^k).

Given unit basis vectors u_i ⊗ u_j ∈ V ⊗ V, let (k_1, k_2, k_3) ∈ℤ_≥ 0^3 such that

    ⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩≠ 0.

F_3, F_2, F_1 may need to be in a different order, but the values of k= (k_1, k_2, k_3) should be the same. 

Set G: V ⊗ V → V ⊗ V by 

    G(u_i ⊗ u_j) =⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩ u_i ⊗ u_j

where k is the above term. 

Now, set S = G^-1HG. 

    a^-1S-Id has rows that sum to 0.


    For fixed u_i ⊗ u_j, we have that the sum of the row indexed by u_i ⊗ u_j equals
    
    ∑_w_1,w_2⟨ u_i ⊗ u_j | H | w_1 ⊗ w_2 ⟩⟨ w_1 ⊗ w_2 | G | w_1 ⊗ w_2 ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩

    which then equals
    
    ∑_w_1,w_2[ ⟨ u_i ⊗ u_j | H | w_1 ⊗ w_2 ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩∑_k⟨ w_1 ⊗ w_2 |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩]

    By commutation between the coproduct and H, this must then equal
    
    ∑_k⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1)H | v ⊗ v ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩

    Finally, this equals
    
    ∑_ka⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩ = a

    Multiplying by a^-1 and subtracting the identity, we obtain a matrix whose rows sum to 0.



 §.§ Explicit values of Ei,Fi,qHi in W tensor W


Now we give explicitly the values of π_W ⊗ W(E_i), π_W ⊗ W(F_i), π_W ⊗ W(q^H_i) for 1 ≤ i ≤ 3. When performing the calculations, the decomposition of W⊗ W into irreducible representations and weight spaces significantly reduces the computational effort, allowing all computations to be done by hand. The matrices below are given for references only, and are strictly speaking not necessary for the proof. In the expressions below, E_i,j denotes the 400 × 400 matrix with all entries 0 except for the (i,j)th entry, which is a 1.


    π_W ⊗ W(E_1) =    (q^3+q)(E_1,2+E_2,3+E_8,9+E_19,20+E_141,142+E_142,143+E_148,149+E_159,160+E_341,342
       +E_342,343+E_348,349+E_359,360) 
    
        +   (q^2+1)(E_61,62+E_62,63+E_68,69+E_79,80+E_81,82+E_82,83+E_88,89+E_99,100+E_261,262
       +E_262,263+E_268,269+E_279,280+E_281,282+E_282,283+E_288,289+E_299,300) 
    
        +   (q^2)(E_4,6+E_5,7+E_8,11+E_9,13+E_14,16+E_15,17+E_18,19+E_144,146+E_145,147+E_148,151
       +E_149,153+E_154,156+E_155,157+E_158,159+E_344,346+E_345,347+E_348,351+E_349,353
       +E_354,356+E_355,357+E_358,359)
    
        +   (q)(E_64,66+E_65,67+E_68,71+E_69,73+E_74,76+E_75,77+E_78,79+E_84,86+E_85,87+E_88,91
       +E_89,93+E_94,96+E_95,97+E_98,99+E_264,266+E_265,267+E_268,271+E_269,273+E_274,276
       +E_275,277+E_278,279+E_284,286+E_285,287+E_288,291+E_289,293+E_294,296+E_295,297+E_298,299) 
    
        +   (q+q^-1)( ∑_i=1^40 E_i,i+20 + ∑_i=141^160 E_i,i+20 +∑_361^400E_i,i+20 + E_22,23+E_28,29+E_39,40+E_362,363+E_379,380. 
       +E_162,163+E_168,169+E_179,180+E_181,182+E_182,183+E_188,189+E_199,200+E_201,202
       . +E_202,203+E_208,209+E_219,220+E_221,222+E_222,223+E_228,229+E_239,240+E_361,362) 
    
        +   (1) ( ∑_i=61^100 E_i,i+40 + ∑_i=141^160 E_i,i+60+∑_i=161^180 E_i,i+80 +∑_i=261^300 E_i,i+40 +∑_i=341^360E_i,i+20. 
       +E_24,26+E_25,27+E_28,31+E_29,33+E_34,36+E_35,37+E_38,39
       +E_164,166+E_165,167+E_169,173+E_174,176+E_175,177+E_178,179
       +E_184,186+E_185,187+E_188,191+E_189,193+E_194,196+E_195,197+E_198,199+E_204,206
       +E_205,207+E_208,211+E_209,213+E_214,216+E_215,217+E_218,219+E_224,226+E_225,227
       +E_228,231+E_229,233+E_234,236+E_235,237+E_238,239
       .+E_364,366+E_365,367+E_368,371+E_369,373+E_374,376+E_375,377+E_378,379)) 
    
        +   (q^-1)(E_104,106+E_105,107+E_108,111+E_109,113+E_114,116+E_115,117+E_118,119+E_124,126
       +E_125,127+E_128,131+E_129,133+E_134,136+E_135,137+E_138,139+E_304,306+E_305,307
       +E_308,311+E_309,313+E_314,316+E_315,317+E_318,319+E_324,326+E_325,327+E_328,331
       +E_329,333+E_334,336+E_335,337+E_338,339) 
    
        +   (q^-2)(E_44,46+E_45,47+E_48,51+E_49,53+E_54,56+E_55,57+E_58,59+E_244,246+E_245,247
       +E_248,251+E_249,253+E_254,256+E_255,257+E_258,259+E_384,386+E_385,387+E_388,391
       +E_389,393+E_394,396+E_395,397+E_398,399) 
    
        +   (1+q^-2)(E_101,102+E_102,103+E_108,109+E_119,120+E_121,122+E_122,123+E_128,129+E_139,140
       +E_301,302+E_302,303+E_308,309+E_319,320+E_321,322+E_322,323+E_328,329+E_339,340) 
    
        +   (q^-1+q^-3)(E_41,42+E_42,43+E_48,49+E_59,60+E_241,242+E_242,243+E_248,249+E_259,260
       +E_381,382+E_382,383+E_388,389+E_399,400)



    π_W ⊗ W(E_2)    =
    (q^3 + q)(E_46,50+E_47,51+E_51,54+E_52,55+E_53,56+E_55,58+E_126,130+E_127,131+E_131,134
       +E_132,135+E_133,136+E_135,138+E_226,230+E_227,231+E_231,234+E_232,235+E_233,236+E_235,238)
    
    +   (q^2 + 1)(E_26,30+E_27,31+E_31,34+E_32,35+E_33,36+E_35,38+E_86,90+E_87,91+E_91,94+E_92,95
       +E_93,96+E_95,98+E_246,250+E_247,251+E_251,254+E_252,255+E_253,256+E_255,258+E_326,330
       +E_327,331+E_331,334+E_332,335+E_333,336+E_335,338)
    
    +   (q^2)(E_42,44+E_43,46+E_45,48+E_47,49+E_57,59+E_122,124+E_123,126+E_125,128+E_127,129
       +E_137,139+E_222,224+E_223,226+E_225,228+E_227,229+E_237,239)
    
    +   (q)(E_22,24+E_23,26+E_25,28+E_27,29+E_37,39+E_82,84+E_83,86+E_85,88+E_87,89+E_97,99
       +E_242,244+E_243,246+E_245,248+E_247,249+E_257,259+E_322,324+E_323,326+E_325,328
       +E_327,329+E_337,339)
    
    +   (q + q^-1)(∑_i=101^140 E_i, i+80 +∑_i=201^260  E_i,i+60 + ∑_i=281 E_i,i+60 + E_6,10+E_7,11+E_11,14+E_12,15+E_13,16.
       +E_15,18+E_106,110+E_107,111+E_111,114+E_112,115+E_113,116+E_115,118+E_166,170+E_167,171
       +E_171,174+E_172,175+E_173,176 +E_175,178+E_206,210
    +E_207,211+E_211,214+E_212,215
       +E_213,216+E_215,218+E_286,290+E_287,291+E_291,294+E_292,295+E_293,296+E_295,298
       .+E_386,390+E_387,391+E_391,394+E_392,395+E_393,396+E_395,398)
    
    +   (1)(∑_i=21^40E_i,i+40 +∑_i=41^60E_i,i+60∑_i=81^100E_i,i+60 + ∑_i=121^140E_i,i+40 +∑_i=321^340 E_i,i+40 +E_2,4+E_3,6+E_5,8. 
       +E_7,9+E_17,19+E_102,104+E_103,106+E_105,108+E_107,109+E_117,119+E_162,164+E_163,166
       +E_165,168+E_167,169+E_177,179+E_202,204+E_203,206+E_205,208 +E_207,209+E_217,219+E_282,284
       +E_283,286+E_285,288+E_287,289+E_297,299+E_382,384+E_383,386 +E_385,388+E_387,389+E_397,399)
    
    +   (q^-1)(E_62,64+E_63,66+E_65,68+E_67,69+E_77,79+E_142,144+E_143,146+E_145,148+E_147,149
       +E_157,159+E_302,304+E_303,306+E_305,308+E_307,309+E_317,319+E_362,364+E_363,366
       +E_365,368+E_367,369+E_377,379)
    
    +   (q^-2)(E_182,184+E_183,186+E_185,188+E_187,189+E_197,199+E_262,264+E_263,266+E_265,268
       +E_267,269+E_277,279+E_342,344+E_343,346+E_345,348+E_347,349+E_357,359)
    
    +   (1 + q^-2)(E_66,70+E_67,71+E_71,74+E_72,75+E_73,76+E_75,78+E_146,150+E_147,151+E_151,154
       +E_152,155+E_153,156+E_155,158+E_306,310+E_307,311+E_311,314+E_312,315+E_313,316
       +E_315,318+E_366,370+E_367,371+E_371,374+E_372,375+E_373,376+E_375,378)
    
    +   (q^-1 + q^-3)(E_186,190+E_187,191+E_191,194+E_192,195+E_193,196+E_195,198+E_266,270+E_267,271
       +E_271,274+E_272,275+E_273,276+E_275,278+E_346,350+E_347,351+E_351,354+E_352,355
       +E_353,356+E_355,358)



    π_W ⊗ W(E_3)    =
    (-q^3 - q)(E_46,51+E_47,52+E_50,54+E_51,55+E_53,57+E_54,58+E_106,111+E_107,112+E_110,114
       +E_111,115+E_113,117+E_114,118+E_186,191+E_187,192+E_190,194+E_191,195+E_193,197+E_194,198)
    
    +   (-q^2 - 1)(E_26,31+E_27,32+E_30,34+E_31,35+E_33,37+E_34,38+E_66,71+E_67,72+E_70,74+E_71,75
       +E_73,77+E_74,78+E_246,251+E_247,252+E_250,254+E_251,255+E_253,257+E_254,258+E_306,311
       +E_307,312+E_310,314+E_311,315+E_313,317+E_314,318)
    
    +   (-q^2)(E_42,45+E_43,47+E_44,48+E_46,49+E_56,59+E_102,105+E_103,107+E_104,108+E_106,109
       +E_116,119+E_182,185+E_183,187+E_184,188+E_186,189+E_196,199)
    
    +   (-q)(E_22,25+E_23,27+E_24,28+E_26,29+E_36,39+E_62,65+E_63,67+E_64,68+E_66,69+E_76,79
       +E_242,245+E_243,247+E_244,248+E_246,249+E_256,259+E_302,305+E_303,307+E_304,308
       +E_306,309+E_316,319)
    
    +   (-q - q^-1)(∑_i=101^140E_i,i+100+∑_i=181^220E_i,i+80+ ∑_i=241^280E_i,i+80+E_6,11+E_7,12+E_10,14+E_11,15.
       +E_13,17+E_14,18+E_126,131+E_127,132+E_130,134+E_131,135+E_133,137+E_134,138+E_166,171
       +E_167,172+E_170,174+E_171,175+E_173,177+E_174,178+E_206,211+E_207,212+E_210,214+E_211,215
       +E_213,217+E_214,218+E_267,272+E_270,274+E_271,275+E_273,277+E_274,278+E_386,391+E_387,392
       +E_390,394+E_391,395+E_393,397+E_394,398)
    
    +   (-1)(∑_i=21^40 + ∑_i=41 E_i,i+80+E_i,i+60^80+∑_i=101^120E_i,i+60+∑_i=301^320E_i,i+60+E_2,5+E_3,7+E_4,8+E_6,9. 
       ++E_16,19+E_122,125+E_123,127+E_124,128+E_126,129+E_136,139+E_162,165+E_163,167
       +E_164,168+E_166,169+E_176,179+E_202,205+E_203,207+E_204,208+E_206,209+E_216,219+E_262,265
       +E_263,267+E_264,268+E_266,269+E_276,279+E_382,385+E_383,387+E_384,388+E_386,389+E_396,399)
    
    +   (-q^-1)(E_82,85+E_83,87+E_84,88+E_86,89+E_96,99+E_142,145+E_143,147+E_144,148+E_146,149
       +E_156,159+E_322,325+E_323,327+E_324,328+E_326,329+E_336,339+E_362,365+E_363,367
       +E_364,368+E_366,369+E_376,379)
    
    +   (-q^-2)(E_222,225+E_223,227+E_224,228+E_226,229+E_236,239+E_282,285+E_283,287+E_284,288
       +E_286,289+E_296,299+E_342,345+E_343,347+E_344,348+E_346,349+E_356,359)
    
    +   (-1 - q^-2)(E_86,91+E_87,92+E_90,94+E_91,95+E_93,97+E_94,98+E_146,151+E_147,152+E_150,154
       +E_151,155+E_153,157+E_154,158+E_326,331+E_327,332+E_330,334+E_331,335+E_333,337
       +E_334,338+E_366,371+E_367,372+E_370,374+E_371,375+E_373,377+E_374,378)
    
    +   (-q^-1 - q^-3)(E_226,231+E_227,232+E_230,234+E_231,235+E_233,237+E_234,238+E_286,291+E_287,292
       +E_290,294+E_291,295+E_293,297+E_294,298+E_346,351+E_347,352+E_350,354+E_351,355
       +E_353,357+E_354,358)




    π_W ⊗ W(F_1)    =
    (q^3 + q)(E_243,163+E_253,173+E_260,180+E_363,343+E_373,353+E_380,360)
    
    +   (q^2 + 1)(E_246,166+E_247,167+E_256,176+E_257,177+E_366,346+E_367,347+E_376,356+E_377,357)
    
    +   (q^2)(E_23,3+E_33,13+E_40,20+E_43,23+E_53,33+E_60,40+E_103,63+E_113,73+E_120,80+E_123,83
       +E_133,93+E_140,100+E_163,143+E_173,153+E_180,160+E_243,203+E_253,213+E_260,220+E_303,263
       +E_313,273+E_320,280+E_323,283+E_333,293+E_340,300+E_383,363+E_393,373+E_400,380)
    
    +   (q)(E_26,6+E_27,7+E_36,16+E_37,17+E_46,26+E_47,27+E_56,36+E_57,37+E_106,66+E_107,67+E_116,76
       +E_117,77+E_126,86+E_127,87+E_136,96+E_137,97+E_166,146+E_167,147+E_176,156+E_177,157
       +E_246,206+E_247,207+E_256,216+E_257,217+E_306,266+E_307,267+E_316,276+E_317,277+E_326,286
       +E_327,287+E_336,296+E_337,297+E_386,366+E_387,367+E_396,376+E_397,377)
    
    +   (q + q^-1)(E_13,9+E_19,18+E_33,29+E_39,38+E_53,49+E_59,58+E_73,69+E_79,78+E_93,89+E_99,98
       +E_113,109+E_119,118+E_133,129+E_139,138+E_153,149+E_159,158+E_173,169+E_179,178+E_193,189
       +E_199,198+E_213,209+E_219,218+E_233,229+E_239,238+E_242,162+E_249,169+E_250,170+E_251,171
       +E_252,172+E_253,249+E_259,179+E_259,258+E_273,269+E_279,278+E_293,289+E_299,298+E_313,309
       +E_319,318+E_333,329+E_339,338+E_353,349+E_359,358+E_362,342+E_369,349+E_370,350+E_371,351
       +E_372,352+E_373,369+E_379,359+E_379,378+E_393,389+E_399,398)
    
    +   (1)(E_2,1+E_3,2+E_6,4+E_7,5+E_9,8+E_13,11+E_16,14+E_17,15+E_20,19+E_22,2+E_22,21+E_23,22
       +E_26,24+E_27,25+E_29,9+E_29,28+E_30,10+E_31,11+E_32,12+E_33,31+E_36,34+E_37,35+E_39,19
       +E_40,39+E_42,22+E_42,41+E_43,42+E_46,44+E_47,45+E_49,29+E_49,48+E_50,30+E_51,31+E_52,32
       +E_53,51+E_56,54+E_57,55+E_59,39+E_60,59+E_62,61+E_63,62+E_66,64+E_67,65+E_69,68+E_73,71
       +E_76,74+E_77,75+E_80,79+E_82,81+E_83,82+E_86,84+E_87,85+E_89,88+E_93,91+E_96,94+E_97,95
       +E_100,99+E_102,62+E_102,101+E_103,102+E_106,104+E_107,105+E_109,69+E_109,108+E_110,70
       +E_111,71+E_112,72+E_113,111+E_116,114+E_117,115+E_119,79+E_120,119+E_122,82+E_122,121
       +E_123,122+E_126,124+E_127,125+E_129,89+E_129,128+E_130,90+E_131,91+E_132,92+E_133,131
       +E_136,134+E_137,135+E_139,99+E_140,139+E_142,141+E_143,142+E_146,144+E_147,145+E_149,148
       +E_153,151+E_156,154+E_157,155+E_160,159+E_162,142+E_162,161+E_163,162+E_166,164+E_167,165
       +E_169,149+E_169,168+E_170,150+E_171,151+E_172,152+E_173,171+E_176,174+E_177,175+E_179,159
       +E_180,179+E_182,181+E_183,182+E_186,184+E_187,185+E_189,188+E_193,191+E_196,194+E_197,195
       +E_200,199+E_202,201+E_203,202+E_206,204+E_207,205+E_209,208+E_213,211+E_216,214+E_217,215
       +E_220,219+E_222,221+E_223,222+E_226,224+E_227,225+E_229,228+E_233,231+E_236,234+E_237,235
       +E_240,239+E_242,202+E_242,241+E_243,242+E_246,244+E_247,245+E_249,209+E_249,248+E_250,210
       +E_251,211+E_252,212+E_253,251+E_256,254+E_257,255+E_259,219+E_260,259+E_262,261+E_263,262
       +E_266,264+E_267,265+E_269,268+E_273,271+E_276,274+E_277,275+E_280,279+E_282,281+E_283,282
       +E_286,284+E_287,285+E_289,288+E_293,291+E_296,294+E_297,295+E_300,299+E_302,262+E_302,301
       +E_303,302+E_306,304+E_307,305+E_309,269+E_309,308+E_310,270+E_311,271+E_312,272+E_313,311
       +E_316,314+E_317,315+E_319,279+E_320,319+E_322,282+E_322,321+E_323,322+E_326,324+E_327,325
       +E_329,289+E_329,328+E_330,290+E_331,291+E_332,292+E_333,331+E_336,334+E_337,335+E_339,299
       +E_340,339+E_342,341+E_343,342+E_346,344+E_347,345+E_349,348+E_353,351+E_356,354+E_357,355
       +E_360,359+E_362,361+E_363,362+E_366,364+E_367,365+E_369,368+E_373,371+E_376,374+E_377,375
       +E_380,379+E_382,362+E_382,381+E_383,382+E_386,384+E_387,385+E_389,369+E_389,388+E_390,370
       +E_391,371+E_392,372+E_393,391+E_396,394+E_397,395+E_399,379+E_400,399)
    
    +   (q^-1)(E_24,4+E_25,5+E_34,14+E_35,15+E_44,24+E_45,25+E_54,34+E_55,35+E_104,64+E_105,65
       +E_114,74+E_115,75+E_124,84+E_125,85+E_134,94+E_135,95+E_164,144+E_165,145+E_174,154
       +E_175,155+E_244,204+E_245,205+E_254,214+E_255,215+E_304,264+E_305,265+E_314,274
       +E_315,275+E_324,284+E_325,285+E_334,294+E_335,295+E_384,364+E_385,365+E_394,374+E_395,375)
    
    +   (q^-2)(E_21,1+E_28,8+E_38,18+E_41,21+E_48,28+E_58,38+E_101,61+E_108,68+E_118,78+E_121,81
       +E_128,88+E_138,98+E_161,141+E_168,148+E_178,158+E_241,201+E_248,208+E_258,218+E_301,261
       +E_308,268+E_318,278+E_321,281+E_328,288+E_338,298+E_381,361+E_388,368+E_398,378)
    
    +   (1 + q^-2)(E_244,164+E_245,165+E_254,174+E_255,175+E_364,344+E_365,345+E_374,354+E_375,355)
    
    +   (q^-1 + q^-3)(E_241,161+E_248,168+E_258,178+E_361,341+E_368,348+E_378,358)




    π_W ⊗ W(F_1) =
       (q^3 + q)(E_243,163+E_253,173+E_260,180+E_363,343+E_373,353+E_380,360)
    
    +   (q^2 + 1)(E_246,166+E_247,167+E_256,176+E_257,177+E_366,346+E_367,347+E_376,356+E_377,357)
    
    +   (q^2)(E_23,3+E_33,13+E_40,20+E_43,23+E_53,33+E_60,40+E_103,63+E_113,73+E_120,80+E_123,83
       +E_133,93+E_140,100+E_163,143+E_173,153+E_180,160+E_243,203+E_253,213+E_260,220+E_303,263
       +E_313,273+E_320,280+E_323,283+E_333,293+E_340,300+E_383,363+E_393,373+E_400,380)
    
    +   (q)(∑_i=1^4 E_10i+16,10i-4 + E_10i+17,10i-3. 
       . + E_106,66 + E_107,67 + E_116,76) 
    
    +   (q + q^-1)(∑_i=1^20 E_20i-7,20i-11 + E_20i-1,20i-2. 
       + ∑_i=1^2 E_120i+122,180i-18 + E_120i+129,180i-11 + E_120i+130,180i-10 + E_120i+131,180i-9
       . + E_120i+132,180i-8 + E_120i+139,180i-1) 
    
    +   (1)(∑_i=1^20 E_20i-18,20i-19 + E_20i-17,20i-18 + E_20i-11,20i-12 +E_20i,20i-1 + E_20i-14,20i-16. 
       + E_20i-13,20i-15 + E_20i-7,20i-9 + E_20i-4,20i-6 + E_20i-3,20i-5
       + ∑_i ∈{1,2,8,19} E_20i+2,20i-18 + E_20i+9,20i-11 + E_20i+10,20i-10 + E_20i+11,20i-9 + E_20i+12,20i-8
       + E_20i+19,20i-1
       + ∑_i ∈{1,2,8,11,12} E_20i+82,20i+42 + E_20i+89,20i+49 + E_20i+90,20i+50 + E_20i+91,20i+51 + E_20i+92,20i+52
       . + E_20i+99,20i+59) 
    
    +   (q^-1)(∑_i ∈{1,2,3,4,15,16,37,38} E_10i+14,10i-6 + E_10i+15,10i-5. 
       . + ∑_i ∈{1,2,3,4,15,16,21,22,23,24} E_10i+94,10i+54 + E_10i+95,10i+55) 
    
    +   (q^-2)(E_21,1+E_28,8+E_38,18+E_41,21+E_48,28+E_58,38+E_101,61+E_108,68+E_118,78+E_121,81
       +E_128,88+E_138,98+E_161,141+E_168,148+E_178,158+E_241,201+E_248,208+E_258,218+E_301,261
       +E_308,268+E_318,278+E_321,281+E_328,288+E_338,298+E_381,361+E_388,368+E_398,378)
    
    +   (1 + q^-2)(E_244,164+E_245,165+E_254,174+E_255,175+E_364,344+E_365,345+E_374,354+E_375,355)
    
    +   (q^-1 + q^-3)(E_241,161+E_248,168+E_258,178+E_361,341+E_368,348+E_378,358)




    π_W ⊗ W(F_2)    =
    (q^3 + q)(E_110,50+E_114,54+E_118,58)
    
    +   (q^2 + 1)(E_104,44+E_108,48+E_116,56+E_119,59)
    
    +   (q^2)(E_70,30+E_74,34+E_78,38+E_150,90+E_154,94+E_158,98+E_190,110+E_194,114+E_198,118+E_210,130
       +E_214,134+E_218,138+E_270,210+E_274,214+E_278,218+E_290,230+E_294,234+E_298,238+E_350,290
       +E_354,294+E_358,298+E_370,330+E_374,334+E_378,338)
    
    +   (q)(E_64,24+E_68,28+E_76,36+E_79,39+E_144,84+E_148,88+E_156,96+E_159,99+E_184,104+E_188,108
       +E_196,116+E_199,119+E_204,124+E_208,128+E_216,136+E_219,139+E_264,204+E_268,208+E_276,216
       +E_279,219+E_284,224+E_288,228+E_296,236+E_299,239+E_344,284+E_348,288+E_356,296+E_359,299
       +E_364,324+E_368,328+E_376,336+E_379,339)
    
    +   (q + q^-1)(E_6,3+E_26,23+E_46,43+E_66,63+E_86,83+E_101,41+E_106,46+E_106,103+E_109,49+E_111,51
       +E_115,55+E_120,60+E_126,123+E_146,143+E_166,163+E_186,183+E_206,203+E_226,223+E_246,243
       +E_266,263+E_286,283+E_306,303+E_326,323+E_346,343+E_366,363+E_386,383)
    
    +   (1)(E_4,2+E_8,5+E_10,6+E_11,7+E_14,11+E_15,12+E_18,15+E_19,17+E_24,22+E_28,25+E_30,26
       +E_31,27+E_34,31+E_35,32+E_38,35+E_39,37+E_44,42+E_48,45+E_50,46+E_51,47+E_54,51
       +E_55,52+E_58,55+E_59,57+E_61,21+E_64,62+E_66,26+E_68,65+E_69,29+E_70,66+E_71,31
       +E_71,67+E_74,71+E_75,35+E_75,72+E_78,75+E_79,77+E_80,40+E_84,82+E_88,85+E_90,86
       +E_91,87+E_94,91+E_95,92+E_98,95+E_99,97+E_104,102+E_108,105+E_110,106+E_111,107
       +E_114,111+E_115,112+E_118,115+E_119,117+E_124,122+E_128,125+E_130,126+E_131,127+E_134,131
       +E_135,132+E_138,135+E_139,137+E_141,81+E_144,142+E_146,86+E_148,145+E_149,89+E_150,146
       +E_151,91+E_151,147+E_154,151+E_155,95+E_155,152+E_158,155+E_159,157+E_160,100+E_164,162
       +E_168,165+E_170,166+E_171,167+E_174,171+E_175,172+E_178,175+E_179,177+E_181,101+E_184,182
       +E_186,106+E_188,185+E_189,109+E_190,186+E_191,111+E_191,187+E_194,191+E_195,115+E_195,192
       +E_198,195+E_199,197+E_200,120+E_201,121+E_204,202+E_206,126+E_208,205+E_209,129+E_210,206
       +E_211,131+E_211,207+E_214,211+E_215,135+E_215,212+E_218,215+E_219,217+E_220,140+E_224,222
       +E_228,225+E_230,226+E_231,227+E_234,231+E_235,232+E_238,235+E_239,237+E_244,242+E_248,245
       +E_250,246+E_251,247+E_254,251+E_255,252+E_258,255+E_259,257+E_261,201+E_264,262+E_266,206
       +E_268,265+E_269,209+E_270,266+E_271,211+E_271,267+E_274,271+E_275,215+E_275,272+E_278,275
       +E_279,277+E_280,220+E_281,221+E_284,282+E_286,226+E_288,285+E_289,229+E_290,286+E_291,231
       +E_291,287+E_294,291+E_295,235+E_295,292+E_298,295+E_299,297+E_300,240+E_304,302+E_308,305
       +E_310,306+E_311,307+E_314,311+E_315,312+E_318,315+E_319,317+E_324,322+E_328,325+E_330,326
       +E_331,327+E_334,331+E_335,332+E_338,335+E_339,337+E_341,281+E_344,342+E_346,286+E_348,345
       +E_349,289+E_350,346+E_351,291+E_351,347+E_354,351+E_355,295+E_355,352+E_358,355+E_359,357
       +E_360,300+E_361,321+E_364,362+E_366,326+E_368,365+E_369,329+E_370,366+E_371,331+E_371,367
       +E_374,371+E_375,335+E_375,372+E_378,375+E_379,377+E_380,340+E_384,382+E_388,385+E_390,386
       +E_391,387+E_394,391+E_395,392+E_398,395+E_399,397)
    
    +   (q^-1)(E_62,22+E_65,25+E_73,33+E_77,37+E_142,82+E_145,85+E_153,93+E_157,97+E_182,102+E_185,105
       +E_193,113+E_197,117+E_202,122+E_205,125+E_213,133+E_217,137+E_262,202+E_265,205+E_273,213
       +E_277,217+E_282,222+E_285,225+E_293,233+E_297,237+E_342,282+E_345,285+E_353,293+E_357,297
       +E_362,322+E_365,325+E_373,333+E_377,337)
    
    +   (q^-2)(E_63,23+E_67,27+E_72,32+E_143,83+E_147,87+E_152,92+E_183,103+E_187,107+E_192,112
       +E_203,123+E_207,127+E_212,132+E_263,203+E_267,207+E_272,212+E_283,223+E_287,227
       +E_292,232+E_343,283+E_347,287+E_352,292+E_363,323+E_367,327+E_372,332)
    
    +   (1 + q^-2)(E_102,42+E_105,45+E_113,53+E_117,57)
    
    +   (q^-1 + q^-3)(E_103,43+E_107,47+E_112,52)
    
    +   (q^2/q + q^-1)(E_270,170+E_274,174+E_278,178+E_310,250+E_314,254+E_318,258)
    
    +   (q/q + q^-1)(E_264,164+E_268,168+E_276,176+E_279,179+E_304,244+E_308,248+E_316,256+E_319,259)
    
    +   (1/q + q^-1)(E_14,9+E_16,13+E_34,29+E_36,33+E_54,49+E_56,53+E_74,69+E_76,73+E_94,89+E_96,93
       +E_114,109+E_116,113+E_134,129+E_136,133+E_154,149+E_156,153+E_174,169+E_176,173+E_194,189
       +E_196,193+E_214,209+E_216,213+E_234,229+E_236,233+E_254,249+E_256,253+E_261,161+E_266,166
       +E_269,169+E_271,171+E_274,269+E_275,175+E_276,273+E_280,180+E_294,289+E_296,293+E_301,241
       +E_306,246+E_309,249+E_311,251+E_314,309+E_315,255+E_316,313+E_320,260+E_334,329+E_336,333
       +E_354,349+E_356,353+E_374,369+E_376,373+E_394,389+E_396,393)
    
    +   (q^-1/q + q^-1)(E_262,162+E_265,165+E_273,173+E_277,177+E_302,242+E_305,245+E_313,253+E_317,257)
    
    +   (q^-2/q + q^-1)(E_263,163+E_267,167+E_272,172+E_303,243+E_307,247+E_312,252)




    π_W ⊗ W(F_2) =
       (q^3 + q)(E_110,50+E_114,54+E_118,58)
    
    +   (q^2 + 1)(E_104,44+E_108,48+E_116,56+E_119,59)
    
    +   (q^2)(∑_i ∈{1,16} E_20i+50,20i+10 + E_20i+54,20i+14 + E_20i+58,20i+18.
       +∑_i ∈{4,10,11,14} E_20i+70,20i+10 + E_20i+74,20i+14 + E_20i+78,20i+18
       .+∑_i ∈{5,6} E_20i+90,20i+10 + E_20i+94,20i+14 + E_20i+98,20i+18) 
    
    +   (q)(∑_i ∈{1,16} E_20i+44,20i+4 + E_20i+48,20i+8 + E_20i+56,20i+16 + E_E_20i+59,20i+19.
       +∑_i ∈{4,10,11,14} E_20i+64,20i+4 + E_20i+68,20i+8 + E_20i+76,20i+16 + E_E_20i+79,20i+19
       . + ∑_i ∈{5,6} E_20i+84,20i+4 + E_20i+88,20i+8 + E_20i+96,20i+16 + E_E_20i+99,20i+19)
    
    +   (q+q^-1) ( ∑_i=1^20 E_20i-14,20i-17.
       . + ∑_j ∈{1,6,9,11,15,20} E_j+100,j+40)
    
    +   (1)( ∑_i=1^20 E_20i-16,20i-18 + E_20i-12,20i-15 + E_20i-10,20i-14 + E_20i-9,20i-13 + E_20i-6,20i-9. 
       + E_20i-5,20i-8 + E_20i-2,20i-5 +  E_20i-1,20i-3
       +∑_i ∈{1,16} E_20i+41,20i+1 + E_20i+46,20i+6 + E_20i+49,20i+9 + E_20i+51,20i+11 + E_20i+55,20i+15
       +E_20i+60,20i+20
       +∑_i ∈{4,10,11,14} E_20i+61,20i+1 + E_20i+66,20i+6 + E_20i+69,20i+9 + E_20i+71,20i+11 + E_20i+75,20i+15
       +E_20i+80,20i+20
       +∑_i ∈{5,6} E_20i+81,20i+1 + E_20i+86,20i+6 + E_20i+89,20i+9 + E_20i+91,20i+11 + E_20i+95,20i+15
       .+E_20i+100,20i+20) 
    
    +   (q^-1)( ∑_i ∈{1,16} E_20i+42,20i+2 + E_20i+45,20i+5 + E_20i+53,20i+13 + E_20i+57,20i+17.
       +∑_i ∈{4,10,11,14} E_20i+62,20i+2 + E_20i+65,20i+5 + E_20i+73,20i+13 + E_20i+77,20i+17
       . +∑_i ∈{5,6} E_20i+82,20i+2 + E_20i+85,20i+5 + E_20i+93,20i+13 + E_20i+97,20i+17) 
    
    +   (q^-2)( ∑_i ∈{1,16} E_20i+43,20i+3 + E_20i+47,20i+7 + E_20i+52,20i+12.
       +∑_i ∈{4,10,11,14} E_20i+63,20i+3 + E_20i+67,20i+7 + E_20i+72,20i+12
       . +∑_i ∈{5,6} E_20i+83,20i+3 + E_20i+87,20i+7 + E_20i+92,20i+12) 
    
    +   (1 + q^-2)(E_102,42+E_105,45+E_113,53+E_117,57)
    
    +   (q^-1 + q^-3)(E_103,43+E_107,47+E_112,52)
    
    +   (q^2/q + q^-1)(E_270,170+E_274,174+E_278,178+E_310,250+E_314,254+E_318,258)
    
    +   (q/q + q^-1)(E_264,164+E_268,168+E_276,176+E_279,179+E_304,244+E_308,248+E_316,256+E_319,259)
    
    +   (1/q + q^-1)( ∑_i=1^20 E_20i-6,20i-11 + E_20i-4,20i-7.
       . + ∑_j ∈{1,6,9,11,15,20} E_j+300,j+240 + E_j+260,j+160) 
    
    +   (q^-1/q + q^-1)(E_262,162+E_265,165+E_273,173+E_277,177+E_302,242+E_305,245+E_313,253+E_317,257)
    
    +   (q^-2/q + q^-1)(E_263,163+E_267,167+E_272,172+E_303,243+E_307,247+E_312,252)




    π_W ⊗ W(F_3)    =
    -(q^3 + q)(E_132,52+E_135,55+E_138,58)
    
    -   (q^2 + 1)(E_125,45+E_128,48+E_137,57+E_139,59)
    
    -   (q^2)(E_92,32+E_95,35+E_98,38+E_152,72+E_155,75+E_158,78+E_212,112+E_215,115+E_218,118+E_232,132
       +E_235,135+E_238,138+E_272,192+E_275,195+E_278,198+E_292,212+E_295,215+E_298,218+E_352,272
       +E_355,275+E_358,278+E_372,312+E_375,315+E_378,318)
    
    -   (q)(E_85,25+E_88,28+E_97,37+E_99,39+E_145,65+E_148,68+E_157,77+E_159,79+E_205,105+E_208,108
       +E_217,117+E_219,119+E_225,125+E_228,128+E_237,137+E_239,139+E_265,185+E_268,188+E_277,197
       +E_279,199+E_285,205+E_288,208+E_297,217+E_299,219+E_345,265+E_348,268+E_357,277+E_359,279
       +E_365,305+E_368,308+E_377,317+E_379,319)
    
    -   (q + q^-1)(E_7,3+E_27,23+E_47,43+E_67,63+E_87,83+E_107,103+E_121,41+E_127,47+E_127,123
       +E_129,49+E_131,51+E_134,54+E_140,60+E_147,143+E_167,163+E_187,183+E_207,203
       +E_227,223+E_247,243+E_267,263+E_287,283+E_307,303+E_327,323+E_347,343+E_367,363+E_387,383)
    
    -   (1)(E_5,2+E_8,4+E_11,6+E_12,7+E_14,10+E_15,11+E_18,14+E_19,16+E_25,22+E_28,24+E_31,26
       +E_32,27+E_34,30+E_35,31+E_38,34+E_39,36+E_45,42+E_48,44+E_51,46+E_52,47+E_54,50
       +E_55,51+E_58,54+E_59,56+E_65,62+E_68,64+E_71,66+E_72,67+E_74,70+E_75,71+E_78,74
       +E_79,76+E_81,21+E_85,82+E_87,27+E_88,84+E_89,29+E_91,31+E_91,86+E_92,87+E_94,34
       +E_94,90+E_95,91+E_98,94+E_99,96+E_100,40+E_105,102+E_108,104+E_111,106+E_112,107
       +E_114,110+E_115,111+E_118,114+E_119,116+E_125,122+E_128,124+E_131,126+E_132,127+E_134,130
       +E_135,131+E_138,134+E_139,136+E_141,61+E_145,142+E_147,67+E_148,144+E_149,69+E_151,71
       +E_151,146+E_152,147+E_154,74+E_154,150+E_155,151+E_158,154+E_159,156+E_160,80+E_165,162
       +E_168,164+E_171,166+E_172,167+E_174,170+E_175,171+E_178,174+E_179,176+E_185,182+E_188,184
       +E_191,186+E_192,187+E_194,190+E_195,191+E_198,194+E_199,196+E_201,101+E_205,202+E_207,107
       +E_208,204+E_209,109+E_211,111+E_211,206+E_212,207+E_214,114+E_214,210+E_215,211+E_218,214
       +E_219,216+E_220,120+E_221,121+E_225,222+E_227,127+E_228,224+E_229,129+E_231,131+E_231,226
       +E_232,227+E_234,134+E_234,230+E_235,231+E_238,234+E_239,236+E_240,140+E_245,242+E_248,244
       +E_251,246+E_252,247+E_254,250+E_255,251+E_258,254+E_259,256+E_261,181+E_265,262+E_267,187
       +E_268,264+E_269,189+E_271,191+E_271,266+E_272,267+E_274,194+E_274,270+E_275,271+E_278,274
       +E_279,276+E_280,200+E_281,201+E_285,282+E_287,207+E_288,284+E_289,209+E_291,211+E_291,286
       +E_292,287+E_294,214+E_294,290+E_295,291+E_298,294+E_299,296+E_300,220+E_305,302+E_308,304
       +E_311,306+E_312,307+E_314,310+E_315,311+E_318,314+E_319,316+E_325,322+E_328,324+E_331,326
       +E_332,327+E_334,330+E_335,331+E_338,334+E_339,336+E_341,261+E_345,342+E_347,267+E_348,344
       +E_349,269+E_351,271+E_351,346+E_352,347+E_354,274+E_354,350+E_355,351+E_358,354+E_359,356
       +E_360,280+E_361,301+E_365,362+E_367,307+E_368,364+E_369,309+E_371,311+E_371,366+E_372,367
       +E_374,314+E_374,370+E_375,371+E_378,374+E_379,376+E_380,320+E_385,382+E_388,384+E_391,386
       +E_392,387+E_394,390+E_395,391+E_398,394+E_399,396)
    
    -   (q^-1)(E_82,22+E_84,24+E_93,33+E_96,36+E_142,62+E_144,64+E_153,73+E_156,76+E_202,102+E_204,104
       +E_213,113+E_216,116+E_222,122+E_224,124+E_233,133+E_236,136+E_262,182+E_264,184+E_273,193
       +E_276,196+E_282,202+E_284,204+E_293,213+E_296,216+E_342,262+E_344,264+E_353,273+E_356,276
       +E_362,302+E_364,304+E_373,313+E_376,316)
    
    -   (q^-2)(E_83,23+E_86,26+E_90,30+E_143,63+E_146,66+E_150,70+E_203,103+E_206,106+E_210,110+E_223,123
       +E_226,126+E_230,130+E_263,183+E_266,186+E_270,190+E_283,203+E_286,206+E_290,210+E_343,263
       +E_346,266+E_350,270+E_363,303+E_366,306+E_370,310)
    
    -   (1 + q^-2)(E_122,42+E_124,44+E_133,53+E_136,56)
    
    -   (q^-1 + q^-3)(E_123,43+E_126,46+E_130,50)
    
    -   (q^2/q + q^-1)(E_292,172+E_295,175+E_298,178+E_332,252+E_335,255+E_338,258)
    
    -   (q/q + q^-1)(E_285,165+E_288,168+E_297,177+E_299,179+E_325,245+E_328,248+E_337,257+E_339,259)
    
    -   (1/q + q^-1)(E_15,9+E_17,13+E_35,29+E_37,33+E_55,49+E_57,53+E_75,69+E_77,73+E_95,89+E_97,93
       +E_115,109+E_117,113+E_135,129+E_137,133+E_155,149+E_157,153+E_175,169+E_177,173+E_195,189
       +E_197,193+E_215,209+E_217,213+E_235,229+E_237,233+E_255,249+E_257,253+E_275,269+E_277,273
       +E_281,161+E_287,167+E_289,169+E_291,171+E_294,174+E_295,289+E_297,293+E_300,180+E_315,309
       +E_317,313+E_321,241+E_327,247+E_329,249+E_331,251+E_334,254+E_335,329+E_337,333+E_340,260
       +E_355,349+E_357,353+E_375,369+E_377,373+E_395,389+E_397,393)
    
    -   (q^-1/q + q^-1)(E_282,162+E_284,164+E_293,173+E_296,176+E_322,242+E_324,244+E_333,253+E_336,256)
    
    -   (q^-2/q + q^-1)(E_283,163+E_286,166+E_290,170+E_323,243+E_326,246+E_330,250)




    π_W ⊗ W(F_3) =
    -   (q^3 + q)(E_132,52+E_135,55+E_138,58)
    
    -   (q^2 + 1)(E_125,45+E_128,48+E_137,57+E_139,59)
    
    -   (q^2)( ∑_i ∈{1,15} E_20i+72,20i+12 + E_20i+75,20i+15 + E_20i+78,20i+18.
       + ∑_i ∈{3,9,10,13} E_20i+92,20i+12 + E_20i+95,20i+15 + E_20i+98,20i+18
       .+ ∑_i ∈{5,6} E_20i+112,20i+12 + E_20i+115,20i+15 + E_20i+118,20i+18)
    
    -   (q)( ∑_i ∈{1,15} E_20i+65,20i+5 + E_20i+68,20i+8 + E_20i+77,20i+17 + E_20i+79,20i+19.
       + ∑_i ∈{3,9,10,13} E_20i+85,20i+5 + E_20i+88,20i+8 + E_20i+97,20i+17 + E_20i+99,20i+19
       . + ∑_i ∈{5,6} E_20i+105,20i+5 + E_20i+108,20i+8 + E_20i+117,20i+17 + E_20i+119,20i+19)
    
    -   (q+q^-1)( ∑_i=1^20 E_20i-13,20i-17.
       . + ∑_j ∈{1,7,9,11,14,20} E_j+120,j+40)
       -(1)( ∑_i=1^20 E_20i-15,20i-18 + E_20i-12,20i-16 + E_20i-9,20i-14 + E_20i-8,20i-13 + E_20i-6,20i-10.
        + E_20i-5,20i-9 + E_20i-2,20i-6 + E_20i-1,20i-4
       + ∑_i ∈{1,15} E_20i+61,20i+1 + E_20i+67,20i+7 + E_20i+69,20i+9 + E_20i+71,20i+11 + E_20i+74,20i+14
       + E_20i+80,20i+20
       + ∑_i ∈{3,9,10,13} E_20i+81,20i+1 + E_20i+87,20i+7 + E_20i+89,20i+9 + E_20i+91,20i+11 + E_20i+94,20i+14
       + E_20i+80,20i+20
       + ∑_i ∈{5,6} E_20i+101,20i+1 + E_20i+107,20i+7 + E_20i+109,20i+9 + E_20i+111,20i+11 + E_20i+114,20i+14
       .+ E_20i+120,20i+20)
    
    -   (q^-1)( ∑_i ∈{1,15} E_20i+62,20i+2 + E_20i+64,20i+4 + E_20i+73,20i+13 + E_20i+76,20i+16.
       + ∑_i ∈{3,9,10,13} E_20i+82,20i+2 + E_20i+84,20i+4 + E_20i+93,20i+13 + E_20i+96,20i+16
       . + ∑_i ∈{5,6} E_20i+102,20i+2 + E_20i+104,20i+4 + E_20i+113,20i+13 + E_20i+116,20i+16)
    
    -   (q^-2)( ∑_i ∈{1,15} E_20i+63,20i+3 + E_20i+66,20i+6 + E_20i+70,20i+10.
       + ∑_i ∈{3,9,10,13} E_20i+83,20i+3 + E_20i+86,20i+6 + E_20i+90,20i+10
       . + ∑_i ∈{1,15} E_20i+103,20i+3 + E_20i+106,20i+6 + E_20i+110,20i+10)
    
    -   (1 + q^-2)(E_122,42+E_124,44+E_133,53+E_136,56)
    
    -   (q^-1 + q^-3)(E_123,43+E_126,46+E_130,50)
    
    -   (q^2/q + q^-1)(E_292,172+E_295,175+E_298,178+E_332,252+E_335,255+E_338,258)
    
    -   (q/q + q^-1)(E_285,165+E_288,168+E_297,177+E_299,179+E_325,245+E_328,248+E_337,257+E_339,259)
    
    -   (1/q + q^-1)( ∑_i=1^20 E_20i-5,20i-11 + E_20i-3,20i-7.
       . + ∑_j ∈{1,7,9,11,14,20} E_j+280,j+160 + E_j+320,j+240) 
    
    -   (q^-1/q + q^-1)(E_282,162+E_284,164+E_293,173+E_296,176+E_322,242+E_324,244+E_333,253+E_336,256)
    
    -   (q^-2/q + q^-1)(E_283,163+E_286,166+E_290,170+E_323,243+E_326,246+E_330,250)



    π_W ⊗ W(q^H_1)    =
    (q^4)(E_1,1+E_8,8+E_18,18+E_141,141+E_148,148+E_158,158+E_341,341+E_348,348+E_358,358)
    
    +   (q^3)(E_4,4+E_5,5+E_14,14+E_15,15+E_61,61+E_68,68+E_78,78+E_81,81+E_88,88+E_98,98+E_144,144
       +E_145,145+E_154,154+E_155,155+E_261,261+E_268,268+E_278,278+E_281,281+E_288,288+E_298,298
       +E_344,344+E_345,345+E_354,354+E_355,355)
    
    +   (q^2)(E_2,2+E_9,9+E_10,10+E_11,11+E_12,12+E_19,19+E_21,21+E_28,28+E_38,38+E_64,64+E_65,65
       +E_74,74+E_75,75+E_84,84+E_85,85+E_94,94+E_95,95+E_142,142+E_149,149+E_150,150+E_151,151
       +E_152,152+E_159,159+E_161,161+E_168,168+E_178,178+E_181,181+E_188,188+E_198,198+E_201,201
       +E_208,208+E_218,218+E_221,221+E_228,228+E_238,238+E_264,264+E_265,265+E_274,274+E_275,275
       +E_284,284+E_285,285+E_294,294+E_295,295+E_342,342+E_349,349+E_350,350+E_351,351+E_352,352
       +E_359,359+E_361,361+E_368,368+E_378,378)
    
    +   (q)(E_6,6+E_7,7+E_16,16+E_17,17+E_24,24+E_25,25+E_34,34+E_35,35+E_62,62+E_69,69+E_70,70
       +E_71,71+E_72,72+E_79,79+E_82,82+E_89,89+E_90,90+E_91,91+E_92,92+E_99,99+E_101,101
       +E_108,108+E_118,118+E_121,121+E_128,128+E_138,138+E_146,146+E_147,147+E_156,156+E_157,157
       +E_164,164+E_165,165+E_174,174+E_175,175+E_184,184+E_185,185+E_194,194+E_195,195+E_204,204
       +E_205,205+E_214,214+E_215,215+E_224,224+E_225,225+E_234,234+E_235,235+E_262,262+E_269,269
       +E_270,270+E_271,271+E_272,272+E_279,279+E_282,282+E_289,289+E_290,290+E_291,291+E_292,292
       +E_299,299+E_301,301+E_308,308+E_318,318+E_321,321+E_328,328+E_338,338+E_346,346+E_347,347
       +E_356,356+E_357,357+E_364,364+E_365,365+E_374,374+E_375,375)
    
    +   (1)(E_3,3+E_13,13+E_20,20+E_22,22+E_29,29+E_30,30+E_31,31+E_32,32+E_39,39+E_41,41+E_48,48
       +E_58,58+E_66,66+E_67,67+E_76,76+E_77,77+E_86,86+E_87,87+E_96,96+E_97,97+E_104,104
       +E_105,105+E_114,114+E_115,115+E_124,124+E_125,125+E_134,134+E_135,135+E_143,143+E_153,153
       +E_160,160+E_162,162+E_169,169+E_170,170+E_171,171+E_172,172+E_179,179+E_182,182+E_189,189
       +E_190,190+E_191,191+E_192,192+E_199,199+E_202,202+E_209,209+E_210,210+E_211,211+E_212,212
       +E_219,219+E_222,222+E_229,229+E_230,230+E_231,231+E_232,232+E_239,239+E_241,241+E_248,248
       +E_258,258+E_266,266+E_267,267+E_276,276+E_277,277+E_286,286+E_287,287+E_296,296+E_297,297
       +E_304,304+E_305,305+E_314,314+E_315,315+E_324,324+E_325,325+E_334,334+E_335,335+E_343,343
       +E_353,353+E_360,360+E_362,362+E_369,369+E_370,370+E_371,371+E_372,372+E_379,379+E_381,381
       +E_388,388+E_398,398)
    
    +   (q^-1)(E_26,26+E_27,27+E_36,36+E_37,37+E_44,44+E_45,45+E_54,54+E_55,55+E_63,63+E_73,73
       +E_80,80+E_83,83+E_93,93+E_100,100+E_102,102+E_109,109+E_110,110+E_111,111+E_112,112
       +E_119,119+E_122,122+E_129,129+E_130,130+E_131,131+E_132,132+E_139,139+E_166,166+E_167,167
       +E_176,176+E_177,177+E_186,186+E_187,187+E_196,196+E_197,197+E_206,206+E_207,207+E_216,216
       +E_217,217+E_226,226+E_227,227+E_236,236+E_237,237+E_244,244+E_245,245+E_254,254+E_255,255
       +E_263,263+E_273,273+E_280,280+E_283,283+E_293,293+E_300,300+E_302,302+E_309,309+E_310,310
       +E_311,311+E_312,312+E_319,319+E_322,322+E_329,329+E_330,330+E_331,331+E_332,332+E_339,339
       +E_366,366+E_367,367+E_376,376+E_377,377+E_384,384+E_385,385+E_394,394+E_395,395)
    
    +   (q^-2)(E_23,23+E_33,33+E_40,40+E_42,42+E_49,49+E_50,50+E_51,51+E_52,52+E_59,59+E_106,106
       +E_107,107+E_116,116+E_117,117+E_126,126+E_127,127+E_136,136+E_137,137+E_163,163+E_173,173
       +E_180,180+E_183,183+E_193,193+E_200,200+E_203,203+E_213,213+E_220,220+E_223,223+E_233,233
       +E_240,240+E_242,242+E_249,249+E_250,250+E_251,251+E_252,252+E_259,259+E_306,306+E_307,307
       +E_316,316+E_317,317+E_326,326+E_327,327+E_336,336+E_337,337+E_363,363+E_373,373+E_380,380
       +E_382,382+E_389,389+E_390,390+E_391,391+E_392,392+E_399,399)
    
    +   (q^-3)(E_46,46+E_47,47+E_56,56+E_57,57+E_103,103+E_113,113+E_120,120+E_123,123+E_133,133
       +E_140,140+E_246,246+E_247,247+E_256,256+E_257,257+E_303,303+E_313,313+E_320,320
       +E_323,323+E_333,333+E_340,340+E_386,386+E_387,387+E_396,396+E_397,397)
    
    +   (q^-4)(E_43,43+E_53,53+E_60,60+E_243,243+E_253,253+E_260,260+E_383,383+E_393,393+E_400,400)



    π_W ⊗ W(q^H_2)    =
    (q^4)(E_43,43+E_47,47+E_52,52+E_123,123+E_127,127+E_132,132+E_223,223+E_227,227+E_232,232)
    
    +   (q^3)(E_23,23+E_27,27+E_32,32+E_42,42+E_45,45+E_53,53+E_57,57+E_83,83+E_87,87+E_92,92+E_122,122
       +E_125,125+E_133,133+E_137,137+E_222,222+E_225,225+E_233,233+E_237,237+E_243,243+E_247,247
       +E_252,252+E_323,323+E_327,327+E_332,332)
    
    +   (q^2)(E_3,3+E_7,7+E_12,12+E_22,22+E_25,25+E_33,33+E_37,37+E_41,41+E_46,46+E_49,49+E_51,51
       +E_55,55+E_60,60+E_82,82+E_85,85+E_93,93+E_97,97+E_103,103+E_107,107+E_112,112+E_121,121
       +E_126,126+E_129,129+E_131,131+E_135,135+E_140,140+E_163,163+E_167,167+E_172,172+E_203,203
       +E_207,207+E_212,212+E_221,221+E_226,226+E_229,229+E_231,231+E_235,235+E_240,240+E_242,242
       +E_245,245+E_253,253+E_257,257+E_283,283+E_287,287+E_292,292+E_322,322+E_325,325+E_333,333
       +E_337,337+E_383,383+E_387,387+E_392,392)
    
    +   (q)(E_2,2+E_5,5+E_13,13+E_17,17+E_21,21+E_26,26+E_29,29+E_31,31+E_35,35+E_40,40+E_44,44
       +E_48,48+E_56,56+E_59,59+E_63,63+E_67,67+E_72,72+E_81,81+E_86,86+E_89,89+E_91,91
       +E_95,95+E_100,100+E_102,102+E_105,105+E_113,113+E_117,117+E_124,124+E_128,128+E_136,136
       +E_139,139+E_143,143+E_147,147+E_152,152+E_162,162+E_165,165+E_173,173+E_177,177+E_202,202
       +E_205,205+E_213,213+E_217,217+E_224,224+E_228,228+E_236,236+E_239,239+E_241,241+E_246,246
       +E_249,249+E_251,251+E_255,255+E_260,260+E_282,282+E_285,285+E_293,293+E_297,297+E_303,303
       +E_307,307+E_312,312+E_321,321+E_326,326+E_329,329+E_331,331+E_335,335+E_340,340+E_363,363
       +E_367,367+E_372,372+E_382,382+E_385,385+E_393,393+E_397,397)
    
    +   (1)(E_1,1+E_6,6+E_9,9+E_11,11+E_15,15+E_20,20+E_24,24+E_28,28+E_36,36+E_39,39+E_50,50
       +E_54,54+E_58,58+E_62,62+E_65,65+E_73,73+E_77,77+E_84,84+E_88,88+E_96,96+E_99,99
       +E_101,101+E_106,106+E_109,109+E_111,111+E_115,115+E_120,120+E_130,130+E_134,134+E_138,138
       +E_142,142+E_145,145+E_153,153+E_157,157+E_161,161+E_166,166+E_169,169+E_171,171+E_175,175
       +E_180,180+E_183,183+E_187,187+E_192,192+E_201,201+E_206,206+E_209,209+E_211,211+E_215,215
       +E_220,220+E_230,230+E_234,234+E_238,238+E_244,244+E_248,248+E_256,256+E_259,259+E_263,263
       +E_267,267+E_272,272+E_281,281+E_286,286+E_289,289+E_291,291+E_295,295+E_300,300+E_302,302
       +E_305,305+E_313,313+E_317,317+E_324,324+E_328,328+E_336,336+E_339,339+E_343,343+E_347,347
       +E_352,352+E_362,362+E_365,365+E_373,373+E_377,377+E_381,381+E_386,386+E_389,389+E_391,391
       +E_395,395+E_400,400)
    
    +   (q^-1)(E_4,4+E_8,8+E_16,16+E_19,19+E_30,30+E_34,34+E_38,38+E_61,61+E_66,66+E_69,69+E_71,71
       +E_75,75+E_80,80+E_90,90+E_94,94+E_98,98+E_104,104+E_108,108+E_116,116+E_119,119+E_141,141
       +E_146,146+E_149,149+E_151,151+E_155,155+E_160,160+E_164,164+E_168,168+E_176,176+E_179,179
       +E_182,182+E_185,185+E_193,193+E_197,197+E_204,204+E_208,208+E_216,216+E_219,219+E_250,250
       +E_254,254+E_258,258+E_262,262+E_265,265+E_273,273+E_277,277+E_284,284+E_288,288+E_296,296
       +E_299,299+E_301,301+E_306,306+E_309,309+E_311,311+E_315,315+E_320,320+E_330,330+E_334,334
       +E_338,338+E_342,342+E_345,345+E_353,353+E_357,357+E_361,361+E_366,366+E_369,369+E_371,371
       +E_375,375+E_380,380+E_384,384+E_388,388+E_396,396+E_399,399)
    
    +   (q^-2)(E_10,10+E_14,14+E_18,18+E_64,64+E_68,68+E_76,76+E_79,79+E_110,110+E_114,114+E_118,118
       +E_144,144+E_148,148+E_156,156+E_159,159+E_170,170+E_174,174+E_178,178+E_181,181+E_186,186
       +E_189,189+E_191,191+E_195,195+E_200,200+E_210,210+E_214,214+E_218,218+E_261,261+E_266,266
       +E_269,269+E_271,271+E_275,275+E_280,280+E_290,290+E_294,294+E_298,298+E_304,304+E_308,308
       +E_316,316+E_319,319+E_341,341+E_346,346+E_349,349+E_351,351+E_355,355+E_360,360+E_364,364
       +E_368,368+E_376,376+E_379,379+E_390,390+E_394,394+E_398,398)
    
    +   (q^-3)(E_70,70+E_74,74+E_78,78+E_150,150+E_154,154+E_158,158+E_184,184+E_188,188+E_196,196
       +E_199,199+E_264,264+E_268,268+E_276,276+E_279,279+E_310,310+E_314,314+E_318,318+E_344,344
       +E_348,348+E_356,356+E_359,359+E_370,370+E_374,374+E_378,378)
    
    +   (q^-4)(E_190,190+E_194,194+E_198,198+E_270,270+E_274,274+E_278,278+E_350,350+E_354,354+E_358,358)



    π_W ⊗ W(q^H_3)    =
    (q^4)(E_43,43+E_46,46+E_50,50+E_103,103+E_106,106+E_110,110+E_183,183+E_186,186+E_190,190)
    
    +   (q^3)(E_23,23+E_26,26+E_30,30+E_42,42+E_44,44+E_53,53+E_56,56+E_63,63+E_66,66+E_70,70+E_102,102
       +E_104,104+E_113,113+E_116,116+E_182,182+E_184,184+E_193,193+E_196,196+E_243,243+E_246,246
       +E_250,250+E_303,303+E_306,306+E_310,310)
    
    +   (q^2)(E_3,3+E_6,6+E_10,10+E_22,22+E_24,24+E_33,33+E_36,36+E_41,41+E_47,47+E_49,49+E_51,51
       +E_54,54+E_60,60+E_62,62+E_64,64+E_73,73+E_76,76+E_101,101+E_107,107+E_109,109+E_111,111
       +E_114,114+E_120,120+E_123,123+E_126,126+E_130,130+E_163,163+E_166,166+E_170,170+E_181,181
       +E_187,187+E_189,189+E_191,191+E_194,194+E_200,200+E_203,203+E_206,206+E_210,210+E_242,242
       +E_244,244+E_253,253+E_256,256+E_263,263+E_266,266+E_270,270+E_302,302+E_304,304+E_313,313
       +E_316,316+E_383,383+E_386,386+E_390,390)
    
    +   (q)(E_2,2+E_4,4+E_13,13+E_16,16+E_21,21+E_27,27+E_29,29+E_31,31+E_34,34+E_40,40+E_45,45
       +E_48,48+E_57,57+E_59,59+E_61,61+E_67,67+E_69,69+E_71,71+E_74,74+E_80,80+E_83,83
       +E_86,86+E_90,90+E_105,105+E_108,108+E_117,117+E_119,119+E_122,122+E_124,124+E_133,133
       +E_136,136+E_143,143+E_146,146+E_150,150+E_162,162+E_164,164+E_173,173+E_176,176+E_185,185
       +E_188,188+E_197,197+E_199,199+E_202,202+E_204,204+E_213,213+E_216,216+E_241,241+E_247,247
       +E_249,249+E_251,251+E_254,254+E_260,260+E_262,262+E_264,264+E_273,273+E_276,276+E_301,301
       +E_307,307+E_309,309+E_311,311+E_314,314+E_320,320+E_323,323+E_326,326+E_330,330+E_363,363
       +E_366,366+E_370,370+E_382,382+E_384,384+E_393,393+E_396,396)
    
    +   (1)(E_1,1+E_7,7+E_9,9+E_11,11+E_14,14+E_20,20+E_25,25+E_28,28+E_37,37+E_39,39+E_52,52
       +E_55,55+E_58,58+E_65,65+E_68,68+E_77,77+E_79,79+E_82,82+E_84,84+E_93,93+E_96,96
       +E_112,112+E_115,115+E_118,118+E_121,121+E_127,127+E_129,129+E_131,131+E_134,134+E_140,140
       +E_142,142+E_144,144+E_153,153+E_156,156+E_161,161+E_167,167+E_169,169+E_171,171+E_174,174
       +E_180,180+E_192,192+E_195,195+E_198,198+E_201,201+E_207,207+E_209,209+E_211,211+E_214,214
       +E_220,220+E_223,223+E_226,226+E_230,230+E_245,245+E_248,248+E_257,257+E_259,259+E_261,261
       +E_267,267+E_269,269+E_271,271+E_274,274+E_280,280+E_283,283+E_286,286+E_290,290+E_305,305
       +E_308,308+E_317,317+E_319,319+E_322,322+E_324,324+E_333,333+E_336,336+E_343,343+E_346,346
       +E_350,350+E_362,362+E_364,364+E_373,373+E_376,376+E_381,381+E_387,387+E_389,389+E_391,391
       +E_394,394+E_400,400)
    
    +   (q^-1)(E_5,5+E_8,8+E_17,17+E_19,19+E_32,32+E_35,35+E_38,38+E_72,72+E_75,75+E_78,78+E_81,81
       +E_87,87+E_89,89+E_91,91+E_94,94+E_100,100+E_125,125+E_128,128+E_137,137+E_139,139+E_141,141
       +E_147,147+E_149,149+E_151,151+E_154,154+E_160,160+E_165,165+E_168,168+E_177,177+E_179,179
       +E_205,205+E_208,208+E_217,217+E_219,219+E_222,222+E_224,224+E_233,233+E_236,236+E_252,252
       +E_255,255+E_258,258+E_265,265+E_268,268+E_277,277+E_279,279+E_282,282+E_284,284+E_293,293
       +E_296,296+E_312,312+E_315,315+E_318,318+E_321,321+E_327,327+E_329,329+E_331,331+E_334,334
       +E_340,340+E_342,342+E_344,344+E_353,353+E_356,356+E_361,361+E_367,367+E_369,369+E_371,371
       +E_374,374+E_380,380+E_385,385+E_388,388+E_397,397+E_399,399)
    
    +   (q^-2)(E_12,12+E_15,15+E_18,18+E_85,85+E_88,88+E_97,97+E_99,99+E_132,132+E_135,135+E_138,138
       +E_145,145+E_148,148+E_157,157+E_159,159+E_172,172+E_175,175+E_178,178+E_212,212+E_215,215
       +E_218,218+E_221,221+E_227,227+E_229,229+E_231,231+E_234,234+E_240,240+E_272,272+E_275,275
       +E_278,278+E_281,281+E_287,287+E_289,289+E_291,291+E_294,294+E_300,300+E_325,325+E_328,328
       +E_337,337+E_339,339+E_341,341+E_347,347+E_349,349+E_351,351+E_354,354+E_360,360+E_365,365
       +E_368,368+E_377,377+E_379,379+E_392,392+E_395,395+E_398,398)
    
    +   (q^-3)(E_92,92+E_95,95+E_98,98+E_152,152+E_155,155+E_158,158+E_225,225+E_228,228+E_237,237
       +E_239,239+E_285,285+E_288,288+E_297,297+E_299,299+E_332,332+E_335,335+E_338,338+E_345,345
       +E_348,348+E_357,357+E_359,359+E_372,372+E_375,375+E_378,378)
    
    +   (q^-4)(E_232,232+E_235,235+E_238,238+E_292,292+E_295,295+E_298,298+E_352,352+E_355,355+E_358,358)




 §.§ Listing 4x4 Blocks of pi(W tensor W)(C)


The twenty-four 4 × 4 blocks of π_W ⊗ W(C) take the following forms:


Ten blocks take the form

    [  B_1  B_2  B_3    0;  B_4  B_5  B_3  B_6;  B_7  B_3  B_8  B_9;    0 B_10 B_11 B_12 ]

Four blocks take the form

    [  B_1  B_7  B_6    0; B_13  B_5  B_3  B_3;  B_2  B_3  B_8 B_10;    0  B_9 B_14 B_12 ]

Four blocks take the form

    [ B_15  B_7  B_7  B_3;  B_3 B_16    0 B_10;  B_3    0 B_16 B_10;  B_3  B_6  B_6 B_17 ]

Four blocks take the form

    [ B_15  B_2  B_2  B_3;  B_6 B_16    0  B_9;  B_6    0 B_16  B_9;  B_3  B_3  B_3 B_17 ]

And two blocks take the form

    [  B_1 B_18 B_19    0; B_20  B_5  B_3 B_21; B_22  B_3  B_8 B_23;    0 B_24 B_25 B_12 ]

Where

    B_1    = q^12-q^10+2q^6+2+q^-2-q^-4+2q^-8
    
        B_2    = q^10-2q^8+q^6+q^-4-2q^-6+q^-8
    
        B_3    = q^9-2q^7+q^5+q^-5-2q^-7+q^-9
    
        B_4    = q^12-q^10-q^8+q^6+q^-2-q^-4-q^-6+q^-8
    
        B_5    = 2q^10-2q^8+q^6+q^4+2+2q^-4-2q^-6+q^-8+q^-10
    
        B_6    = q^10-q^8-q^6+q^4+q^-4-q^-6-q^-8+q^-10
    
        B_7    = q^11-q^9-q^7+q^5+q^-3-q^-5-q^-7+q^-9
    
        B_8    = q^10+q^8-2q^6+2q^4+2+q^-4+q^-6-2q^-8+2q^-10
    
        B_9    = q^9-q^7-q^5+q^3+q^-5-q^-7-q^-9+q^-11
    
        B_10   = q^8-2q^6+q^4+q^-6-2q^-8+q^-10
    
        B_11   = q^7-2q^5+q^3+q^-7-2q^-9+q^-11
    
        B_12   = 2q^8-q^4+q^2+2+2q^-6-q^-10+q^-12
    
        B_13   = q^11-2q^9+q^7+q^-3-2q^-5+q^-7
    
        B_14   = q^8-q^6-q^4+q^2+q^-6-q^-8-q^-10+q^-12
    
        B_15   = q^12-2q^8+3q^6+2+q^-2-2q^-6+3q^-8
    
        B_16   = q^10+q^4+2+q^-4+q^-10
    
        B_17   = 3q^8-2q^6+q^2+2+3q^-6-2q^-8+q^-12
    
        B_18   = q^9-3q^7+4q^5-4q^3+4q-4q^-4/q^--3q^-5+q^-7
    
        B_19   = q^8-3q^6+4q^4-4q^2+4-4q^-2+4q^-4-3q^-6+q^-8
    
        B_20   = q^13-2q^9+q^5+q^-2/q^-+q^-9
    
        B_21   = q^11-2q^7+q^3+q^-3-2q^-7+q^-11
    
        B_22   = q^12-2q^8+q^4+q^-2-2q^-6+q^-10
    
        B_23   = q^10-2q^6+q^2+q^-4-2q^-8+q^-12
    
        B_24   = q^7-3q^5+4q^3-4q+4q^-4/q^-+4q^-5-3q^-7+q^-9
    
        B_25   = q^6-3q^4+4q^2-4+4q^-2-4q^-4+4q^-6-3q^-8+q^-10










