# Set-theoretic Yang-Baxter equation, braces and Drinfeld twists

Anastasia Doikou  Dept of Mathematics, Heriot-Watt University, Edinburgh EH14
4AS, and Maxwell Institute for Mathematical Sciences, Edinburgh
<EMAIL_ADDRESS>
###### Abstract.

We consider involutive, non-degenerate, finite set-theoretic solutions of the
Yang-Baxter equation. Such solutions can be always obtained using certain
algebraic structures that generalize nilpotent rings called braces. Our main
aim here is to express such solutions in terms of admissible Drinfeld twists
substantially extending recent preliminary results. We first identify the
generic form of the twists associated to set-theoretic solutions and we show
that these twists are admissible, i.e. they satisfy a certain co-cycle
condition. These findings are also valid for Baxterized solutions of the Yang-
Baxter equation constructed from the set-theoretical ones.

###### Key words and phrases:

Yang-Baxter equation, braces, quantum groups, Drinfeld twists

##  1\. Introduction

The idea of set-theoretic solutions of the Yang-Baxter equation (YBE [3]) was
first suggested by Drinfield [10] and since then there has been a considerable
research activity on this topic from the algebraic point of view [20, 13, 14],
but also in the context of classical integrable systems [1, 31, 26]. More
recently, an algebraic structure that generalizes nilpotent rings, called a
brace, was introduced [27, 28], to describe all finite, involutive, non-
degenerate set-theoretic solutions of the Yang-Baxter equation. Rump showed
[27, 28] that every brace yields a solution to the Yang-Baxter equation, and
every non-degenerate, involutive set-theoretic solution of the Yang-Baxter
equation can be obtained from a brace. Subsequently skew-braces were developed
in [19] to describe non-involutive solutions. This emerging research area has
been particularly fruitful and numerous relevant studies have been produced
over the past few years (see for instance [2, 4, 5], [16]-[18], [19, 21, 29,
30]).

In [7, 8] key links between set-theoretical solutions coming from braces and
quantum integrable systems and the associated quantum algebras were uncovered.
More precisely:

  1. (1)

Quantum groups associated to Baxterized solutions of the Yang-Baxter equations
coming from braces were derived via the FRT construction [15].

  2. (2)

Novel classes of quantum discrete integrable systems with periodic and open
boundary conditions were produced.

  3. (3)

Symmetries of the periodic and open transfer matrices of the novel integrable
systems were identified.

  4. (4)

Preliminary findings on Drinfeld twists for set-theoretic solutions were
presented.

Note that in [13] Etingof, Schedler and Soloviev constructed quantum groups
associated to set-theoretic solutions, however in [7] a different construction
is used coming from parameter dependent solutions of the Yang-Baxter equation
and thus the corresponding quantum groups differ from those in [13].

Our main focus here is the study of involutive set-theoretic solutions coming
from braces, based on the findings presented in [7, 8]. More precisely, our
aim is to identify the explicit form of so called admissible Drinfeld twists
for brace solutions of the YBE. We note that in Drinfeld’s original works on
quantum algebras [11, 12] it was required that the universal ${\cal
R}$-matrix, solution of the Yang-Baxter equation has a semi-classical limit,
i.e. it can be expressed as formal series expansion, ${\cal R}={\bf
1}+\sum_{n=1}^{\infty}h^{n}{\cal R}^{(n)}$, where $h$ is some “deformation”
parameter. In [12] Drinfeld introduced the notion of the twist
${\mathfrak{F}}$, which links distinct solutions of the Yang-Baxter equation,
and consequently yields distinct quasi triangular Hopf or (quasi) Hopf
algebras, provided that the twist is admissible. Such twists have also semi-
classical limits [12], i.e. they can be expressed as formal series expansions
in powers of the deformation parameter, with the leading term being the
identity.

In the analysis of [7, 8] Baxterized $R$-matrices coming from set-theoretic
solutions of the Yang-Baxter equation were identified, being of the form
$R(\lambda)=r+{1\over\lambda}{\cal P}$, where $r$ is the set-theoretic
solution of the Yang-Baxter equation and ${\cal P}$ is the permutation
operator. Interestingly the $r$-matrix does not contain any free parameter
(deformation parameter), and consequently the $R$-matrix has no semi-classical
analogue. This is the pivotal difference between our study here and Drinfeld’s
analysis [12]. A similar observation can be made about the associated Drinfeld
twist, which is explicitly identified in the present study. To be more precise
the main results of this investigation, presented in section 3, are:

  1. (1)

The identification of the explicit form of “local” (2-site) and “global”
($n$-site) Drinfeld twists.

  2. (2)

The proof of the admissibility of Drinfeld’s twists associated to involutive,
non-degenerate, set-theoretic solutions of the Yang-Baxter equation.

### The outline of the paper

We describe below what is achieved in each of the subsequent sections.

  * •

In section 2 we present necessary preliminary notions regarding the set-
theoretic solutions of the YBE and the associated quantum groups derived via
the FRT construction [7]. More specifically:

    1. (1)

In subsection 2.1 we recall the definition of the set-theoretic YBE. We also
recall Rump’s theorem, which states that every involutive, non-degenerate,
set-theoretic solution of the YBE comes from a brace and vice versa.

    2. (2)

In subsection 2.2 we review the construction of the quantum group associated
to involutive, set-theoretic solutions by means of the FRT formulation.

  * •

In section 3 we first recall basic definitions and results on Hopf algebras
and Drinfeld twists [11, 12]. We then move on with the presentation of our
main findings regarding the derivation of admissible Drinfeld twists
associated to set-theoretic solutions of the YBE. Specifically:

    1. (1)

In subsection 3.1 we review basic background on bialgebras, Hopf and quasi-
triangular Hopf algebras. We then introduce the notion of admissible Drinfeld
twist and recall fundamental propositions on admissible twists adjusted for
your purposes here.

    2. (2)

In subsection 3.2 we present the new findings on the Drinfeld twists for
involutive, non-degenerate, set-theoretic solutions. To achieve this we rely
on the results of [8], where the set-theoretic braid $r$-matrix was obtained
from the permutation operator via a similarity transformation. We use the
similarity transformation to derive explicit forms for the set-theoretic
twist. We also derive co-product expressions of the twist, which enable the
derivation of the $n$-twist. From the derived explicit expressions we are able
to show the admissibility of the twist i.e. we show that it satisfies the co-
cycle condition. Note that we introduce the term quasi-admissible twist given
that we only show the admissibility of the set-theoretic twists, without
exploring the notions of the co-unit and antipode, although some related
comments are presented at the end of subsection 3.3.

    3. (3)

In subsection 3.3 we focus on a simple, but characteristic example of set-
theoretic solution of the YBE known as Lyubashenko’s solution [10]). We first
introduce this class of solutions and we show that they can be expressed as
Reshetikhin type twists recalling the results of [8]. We then move on to show
that this is an admissible twist and we derive simple expressions for the
$n$-twist. At the end of this subsection we present some preliminary
observations related to the action of the co-unit on the twisted co-products.
We employ the Lyubashenko solution to illustrate the non-trivial action of the
co-unit.

##  2\. Preliminaries

We present in this section basic background information regarding set-
theoretic solutions of the Yang-Baxter equation and braces as well as a brief
review on the recent findings of [7] on the links between set-theoretic
solutions of the Yang-Baxter equation from braces and quantum algebras.

###  2.1. The set-theoretic Yang-Baxter equation

Let $X=\\{{\mathrm{x}}_{1},{\mathrm{x}}_{2},\ldots,{\mathrm{x}}_{\cal N}\\}$
be a set and ${\check{r}}:X\times X\rightarrow X\times X,$ such that

${\check{r}}(x,y)=\big{(}\sigma_{x}(y),\tau_{y}(x)\big{)}.$

We say that $\check{r}$ is non-degenerate if $\sigma_{x}$ and $\tau_{y}$ are
bijective functions. Also, the solutions $(X,\check{r})$ is involutive:
$\check{r}(\sigma_{x}(y),\tau_{y}(x))=(x,y)$,
($\check{r}\check{r}(x,y)=(x,y)$). We focus on non-degenerate, involutive
solutions of the set-theoretic braid equation:

$({\check{r}}\times id_{X})(id_{X}\times{\check{r}})({\check{r}}\times
id_{X})=(id_{X}\times{\check{r}})({\check{r}}\times
id_{X})(id_{X}\times{\check{r}}).$

Let $V$ be the space of dimension equal to the cardinality of $X$, and with a
slight abuse of notation, let $\check{r}$ also denote the $R$-matrix
associated to the linearisation of ${\check{r}}$ on $V={\mathbb{C}}X$ (see
[30] for more details), i.e. $\check{r}$ is the
${\mathcal{N}}^{2}\times{\mathcal{N}}^{2}$ matrix:

(2.1) $\check{r}=\sum_{x,y,z,w\in X}\check{r}(x,z|y,w)e_{x,z}\otimes e_{y,w},$

where $e_{x,y}$ is the ${\mathcal{N}}\times{\mathcal{N}}$ matrix:
$(e_{x,y})_{z,w}=\delta_{x,z}\delta_{y,w}$. Then for the $\check{r}$-matrix
related to $(X,{\check{r}})$:
${\check{r}}(x,z|y,w)=\delta_{z,\sigma_{x}(y)}\delta_{w,\tau_{y}(x)}$. Notice
that the matrix $\check{r}:V\otimes V\rightarrow V\otimes V$ satisfies the
(constant) Braid equation:

$({\check{r}}\otimes I_{V})(I_{V}\otimes{\check{r}})({\check{r}}\otimes
I_{V})=(I_{V}\otimes{\check{r}})({\check{r}}\otimes
I_{V})(I_{V}\otimes{\check{r}}).$

Notice also that ${\check{r}}^{2}=I_{V\otimes V}$ the identity matrix, because
$\check{r}$ is involutive.

For set-theoretic solutions it is thus convenient to use the matrix notation:

(2.2) ${\check{r}}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes
e_{y,\tau_{y}(x)}.$

Define also, $r={\mathcal{P}}{\check{r}}$, where ${\mathcal{P}}=\sum_{x,y\in
X}e_{x,y}\otimes e_{y,x}$ is the permutation operator, consequently
${r}=\sum_{x,y\in X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}.$ The Yangian
[32] is a special case: $\check{r}(x,z|y,w)=\delta_{z,y}\delta_{w,x}$.

Let us now recall the role of braces in the derivation of set-theoretic
solutions of the Yang-Baxter equation. In [27, 28] Rump showed that every
solution $(X,\check{r})$ can be in a good way embedded in a brace.

######  Definition 2.1 (Proposition $4$, [28]).

A left brace  is an abelian group $(A;+)$ together with a multiplication
$\cdot$ such that the circle operation $a\circ b=a\cdot b+a+b$ makes $A$ into
a group, and $a\cdot(b+c)=a\cdot b+a\cdot c$.

In many papers, an equivalent definition is used [5].

The additive identity of a brace $A$ will be denoted by $0$ and the
multiplicative identity by $1$. In every brace $0=1$. The same notation will
be used for skew braces (in every skew brace $0=1$).

Throughout this paper we will use the following result, which is implicit in
[27, 28] and explicit in Theorem 4.4 of [5].

######  Theorem 2.2.

(Rump’s theorem, [27, 28, 5]). It is known that for an involutive, non-
degenerate solution of the braid equation there is always an underlying brace
$(B,\circ,+)$, such that the maps $\sigma_{x}$ and $\tau_{y}$ come from this
brace, and $X$ is a subset in this brace such that
${\check{r}}(X,X)\subseteq(X,X)$ and
${\check{r}}(x,y)=(\sigma_{x}(y),\tau_{y}(x))$, where $\sigma_{x}(y)=x\circ
y-x$, $\tau_{y}(x)=t\circ x-t$, where $t$ is the inverse of $\sigma_{x}(y)$ in
the circle group $(B,\circ)$. Moreover, we can assume that every element from
$B$ belongs to the additive group $(X,+)$ generated by elements of $X$. In
addition every solution of this type is a non-degenerate, involutive set-
theoretic solution of the braid equation.

We will call the brace $B$ an underlying brace of the solution
$(X,{\check{r}})$, or a brace associated to the solution $(X,{\check{r}})$. We
will also say that the solution $(X,\check{r})$ is associated to brace $B$.
Notice that this is also related to the formula of set-theoretic solutions
associated to the braided group (see [13] and [17]).

###  2.2. Yang-Baxter equation $\&$ quantum groups

In this subsection we briefly review the main results reported in [7] on the
various links between braces, representations of the $A$-type Hecke algebras,
and quantum algebras.

Recall first the Yang-Baxter equation in the braid form in the presence of
spectral parameters $\lambda_{1},\ \lambda_{2}$
($\delta=\lambda_{1}-\lambda_{2}$):

(2.3) $\check{R}_{12}(\delta)\ \check{R}_{23}(\lambda_{1})\
\check{R}_{12}(\lambda_{2})=\check{R}_{23}(\lambda_{2})\
\check{R}_{12}(\lambda_{1})\ \check{R}_{23}(\delta).$

where $\check{R}:V\to V,$ and let in general $\check{R}=\sum_{j}a_{j}\otimes
b_{j},$ then in the index notation $\check{R}_{12}=\sum_{j}a_{j}\otimes
b_{j}\otimes I_{V},$ $\check{R}_{23}=\sum_{j}I_{V}\otimes a_{j}\otimes b_{j}$
and $\check{R}_{13}=\sum_{j}a_{j}\otimes I_{V}\otimes b_{j}.$

We focus here on brace solutions of the YBE, given by (2.2). The brace
solution111All, finite, non-degenerate, involutive, set-theoretic solutions of
the YBE are coming from braces, therefore we will call such solutions brace
solutions. $\check{r}$ (2.2) is a representation of the $A$-type Hecke algebra
for $q=1$ (see also [7]), as $\check{r}$ satisfies the braid relations and
also $\check{r}^{2}=I_{X\otimes X}.$ Due to the fact that the set-theoretic
$\check{r}$ provides a representation of the $A$-type Hecke algebra Baxterized
solutions of the Yang-Baxter equation can be derived [7]:

(2.4) $\check{R}(\lambda)=\lambda\check{r}+I_{X}\otimes I_{X},$

where $I_{X}$ is the identity matrix of dimension equal to the cardinality of
the set $X$.

Let also $R={\cal P}\check{R}$, (recall the permutation operator ${\cal
P}=\sum_{x,y}e_{x,y}\otimes e_{y,x}$), then the following basic properties for
$R$ matrices coming from braces were shown in [7]:

Basic Properties. The brace $R$-matrix satisfies the following fundamental
properties:

(2.5) $\displaystyle R_{12}(\lambda)\
R_{21}(-\lambda)=(-\lambda^{2}+1)I_{X\otimes
X},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{{\it Unitarity}}$ (2.6)
$\displaystyle R_{12}^{t_{1}}(\lambda)\ R_{12}^{t_{2}}(-\lambda-{\cal
N})=\lambda(-\lambda-{\cal N})I_{X\otimes X},~{}~{}~{}~{}~{}\mbox{{\it
Crossing-unitarity}}$ $\displaystyle
R_{12}^{t_{1}t_{2}}(\lambda)=R_{21}(\lambda),$

where ${}^{t_{1,2}}$ denotes transposition on the first, second space
respectively.

The quantum algebra. Our approach on deriving the quantum groups associated to
set-theoretic solutions [7, 8] is based on the FRT construction [15], which is
somehow dual to the Hopf algebraic description [11]. Indeed, the FRT
construction can be considered as an inverse procedure of the one that uses
the quasi-triangular Hopf algebra axioms in obtaining a solution of the YBE.
The FRT construction considers a solution $R:V\otimes V\to V\otimes V$ ($V$ is
usually a finite vector space) of the YBE as an input and produces a bialgebra
as an output.

Given a solution of the Yang-Baxter equation, the quantum algebra is defined
via the fundamental relation [15] (we have multiplied the familiar RTT
relation with the permutation operator):

(2.7) $\check{R}_{12}(\lambda_{1}-\lambda_{2})\ L_{1}(\lambda_{1})\
L_{2}(\lambda_{2})=L_{1}(\lambda_{2})\ L_{2}(\lambda_{1})\
\check{R}_{12}(\lambda_{1}-\lambda_{2}).$

$\check{R}(\lambda)\in\mbox{End}({\mathbb{C}}^{\cal
N})\otimes\mbox{End}({\mathbb{C}}^{\cal N})$, $\
L(\lambda)\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes{\mathfrak{A}}$, where
${\mathfrak{A}}$ is the quantum algebra defined by (2.7). We focus on
solutions associated to braces given by (2.4), (2.2). The defining relations
of the corresponding quantum algebra were derived in [7].

The quantum algebra associated to the brace $R$ matrix (2.4), (2.2) is defined
by generators $L^{(m)}_{zw},\ z,w\in X$, and defining relations

(2.8) $\displaystyle
L_{z,w}^{(n)}L_{\hat{z},\hat{w}}^{(m)}-L_{z,w}^{(m)}L_{\hat{z},\hat{w}}^{(n)}$
$\displaystyle=$ $\displaystyle
L^{(m)}_{z,\sigma_{w}(\hat{w})}L^{(n+1)}_{\hat{z},\tau_{\hat{w}}(w)}-L^{(m+1)}_{z,\sigma_{w}(\hat{w})}L^{(n)}_{\hat{z},\tau_{\hat{w}}(w)}$
$\displaystyle-$ $\displaystyle
L^{(n+1)}_{\sigma_{z}(\hat{z}),w}L^{(m)}_{\tau_{\hat{z}}(z),\hat{w}}+L^{(n)}_{\sigma_{z}(\hat{z},)w}L^{(m+1)}_{\tau_{\hat{z}}(z),\hat{w}}.$

The proof is based on the fundamental relation (2.7) and the form of the brace
$R$\- matrix (for the detailed proof see [7]). Recall also that in the index
notation we define $\check{R}_{12}=\check{R}\otimes\mbox{id}_{\mathfrak{A}}$:

(2.9) $\displaystyle L_{1}(\lambda)=\sum_{z,w\in X}e_{z,w}\otimes I\otimes
L_{z,w}(\lambda),\ \quad L_{2}(\lambda)=\sum_{z,w\in X}I\otimes e_{z,w}\otimes
L_{z,w}(\lambda)$

where $L_{z,w}(\lambda)=\sum_{n=0}^{\infty}\lambda^{-n}L_{z,w}^{(n)}$ and
$L_{z,w}^{(n)}$ are the generators of the affine algebra ${\mathfrak{A}}$ and
$\check{R}$ is given in (2.4), (2.2). The quantum algebra is equipped with a
co-product $\Delta:{\mathfrak{A}}\to{\mathfrak{A}}\otimes{\mathfrak{A}}$ [15,
11]. Indeed, we define
${\mathrm{T}}_{1,23}(\lambda)=L_{13}(\lambda)L_{12}(\lambda),$ which satisfies
(2.7) and is expressed as ${\mathrm{T}}_{1,23}(\lambda)=\sum_{x,y\in
X}e_{x,y}\otimes\Delta(L_{x,y}(\lambda)).$

##  3\. Set-theoretic solutions as Drinfeld twists

In this section we recall basic definitions and results on Hopf algebras and
Drinfeld twists [11, 12], we then move on with the presentation of our main
findings regarding the derivation of admissible Drinfeld twists associated to
set-theoretic solutions of the YBE.

###  3.1. Quasi-triangular Hopf algebras $\&$ Drinfeld twists

Before we present our brief review on Drinfeld’s twists let us first recall
the notion of a quasi-triangular Hopf algebra. The known quantum algebras
defined by the RTT relation (2.7) are quasi-triangular Hopf algebras (on a
detailed discussion on bialgebras, Hopf algebras, and quasi-triangular (quasi)
Hopf algebras we refer the interested reader for instance in [11, 12, 6, 25,
24]). We give below some useful definitions regarding, bialgebras, Hopf and
quasi-triangular Hopf algebras (see also for instance [6, 25]).

######  Definition 3.1.

A bialgebra $H$ is a vector space over some field $k$ with linear maps:

  1. (1)

$m:H\otimes H\to H$, multiplication, $m(a\otimes b)=ab$, which is associative,
i.e. $(ab)c=a(bc)$ $\forall a,b,c\in H.$

  2. (2)

$\eta:k\to H$ such that it produces the unit element for $H$, $\eta(1)={\bf
1}_{H}$ ($\eta(c)=c\cdot{\bf 1}_{H}$).

  3. (3)

$\Delta:H\to H\otimes H$, coproduct $\Delta(a)=\sum_{j}a_{j}\otimes b_{j},$
which is coassociative, i.e.
$(\Delta\otimes\mbox{id})\Delta=(\mbox{id}\otimes\Delta)\Delta.$

  4. (4)

$\epsilon:H\to k$, counit such that
$(\epsilon\otimes\mbox{id})\Delta(a)=(\mbox{id}\otimes\epsilon)\Delta(a)=a$,
$\forall a\in H$.

  5. (5)

$\Delta,\ \epsilon$ are algebra homomorphisms and $H\otimes H$ has the
structure of a tensor product algebra: $(a\otimes b)(c\otimes d)=ac\otimes
bd$, $\forall a,b,c,d\in H.$

In other words, $(H,\ m,\ \eta)$ is an associative algebra and
$(H,\Delta,\epsilon)$ is an associative coalgebra. The compatibility of the
bialgebra axioms is typically represented by commutative diagrams (see for
instance [6, 25] and references therein).

######  Definition 3.2.

A Hopf algebra $H$ is a bialgebra equipped with a bijective linear map, the
“antipode”, $s:H\to H,$ such that
$m\big{(}(s\otimes\mbox{id})\Delta(g)\big{)}=m\big{(}(\mbox{id}\otimes
s)\Delta(g)\big{)}=\epsilon(g)\cdot{\bf 1}_{H},$ $\forall g\in H.$

We may now give the definition of a quasi-triangular Hopf algebra, which is
our main focus in this subsection.

######  Definition 3.3.

Let ${\mathfrak{A}}$ be a Hopf algebra over some field $k$, then
${\mathfrak{A}}$ is a quasi-triangular Hopf algebra if there exists an
invertible element ${\cal R}\in{\cal A}\otimes{\cal A}$:

  1. (1)

${\cal R}\ \Delta(a)=\Delta^{op}(a)\ {\cal R}$, $\ \forall
a\in{\mathfrak{A}}$,
where $\Delta:{\mathfrak{A}}\to{\mathfrak{A}}\otimes{\mathfrak{A}}$ is the co-
product on ${\mathfrak{A}}$ and $\Delta^{op}(a)=\pi\circ\Delta(a)$,
$\pi:{\mathfrak{A}}\to{\mathfrak{A}}$ such that $\pi(a\otimes b)=b\otimes a$.

  2. (2)

$(\mbox{id}\otimes\Delta){\cal R}={\cal R}_{13}{\cal R}_{12},$ and
$(\Delta\otimes\mbox{id}){\cal R}={\cal R}_{13}{\cal R}_{23}$.

Also, the following statements hold:

  * •

The antipode $s:{\mathfrak{A}}\to{\mathfrak{A}}$ satisfies $(\mbox{id}\otimes
s){\cal R}^{-1}={\cal R}$, $(s\otimes\mbox{id}){\cal R}={\cal R}^{-1}$,

  * •

The co-unit $\epsilon:{\mathfrak{A}}\to k$ satisfies
$(\mbox{id}\otimes\epsilon){\cal R}=(\epsilon\otimes\mbox{id}){\cal R}={\bf
1}$.

  * •

Due to (1) and (2) of Definition 3.3 the ${\cal R}$-matrix satisfies the Yang-
Baxter equation ${\cal R}_{12}{\cal R}_{13}{\cal R}_{23}={\cal R}_{23}{\cal
R}_{13}{\cal R}_{12}$.

Proofs of the above statements can be found for instance in [6, 25]. Note that
a sufficient condition for the co-associativity of $\Delta^{(op)}$ is $({\cal
R}\otimes{\bf 1})\ (\Delta\otimes\mbox{id}){\cal R}=({\bf 1}\otimes{\cal R})\
(\mbox{id}\otimes\Delta){\cal R}$, i.e. ${\cal R}$ should satisfy the co-cycle
condition. The co-cycle condition together with (2) of Definition 3.3 lead to
YBE.

The index notation holds in this case as follows. Let ${\cal
R}=\sum_{j}a_{j}\otimes b_{j}\in{\mathfrak{A}}\otimes{\mathfrak{A}}$ then
${\cal R}_{12}=\sum_{j}a_{j}\otimes b_{j}\otimes{\bf 1}$, $\ {\cal
R}_{23}=\sum_{j}{\bf 1}\otimes a_{j}\otimes b_{j}$ etc.

If in addition to the above conditions ${\cal R}{\cal R}^{(op)}={\bf 1}_{{\cal
A}\otimes{\cal A}}$, where ${\cal R}^{(op)}=\sigma\circ{\cal R}$, (or in the
index notation ${\cal R}_{12}{\cal R}_{21}={\bf 1}$), then ${\mathfrak{A}}$ is
a triangular Hopf-algebra

### Drinfelds twists

In [11, 12] Drinfeld introduced the notion of twisting (or deformation) for
(quasi)-Hopf algebras and quasi-triangular (quasi)-Hopf algebras. Recall that
we focus in this subsection on quasi-triangular Hopf algebras. Let
${\mathfrak{A}}$, equipped with $(m,\eta,\epsilon,\Delta,\ s,\ \epsilon)$, be
a quasi-triangular Hopf algebra over ${\mathbb{C}}$ and
${\mathfrak{F}}\in{\mathfrak{A}}\otimes{\mathfrak{A}}$ be an invertible
element such that
$(\mbox{id}\otimes\epsilon){\mathfrak{F}}=(\epsilon\otimes\mbox{id}){\mathfrak{F}}={\bf
1}$. The twist of ${\mathfrak{A}}$ generated by ${\mathfrak{F}}$ defines a new
quasi-triangular Hopf algebra $\tilde{\mathfrak{A}}$, equipped with
$(m,\eta,\epsilon,\tilde{\Delta},\ \tilde{s},\ \tilde{\epsilon}),$ having the
same elements and product law as ${\mathfrak{A}}$, and such that:

(3.1) $\tilde{\cal R}={\mathfrak{F}}^{op}{\cal R}\ {\mathfrak{F}}^{-1},$

where we define ${\mathfrak{F}}^{op}=\pi\circ{\mathfrak{F}}$, and the
permutation (flip) map $\pi:a\otimes b\mapsto b\otimes a$. In the index
notation ${\mathfrak{F}}_{12}^{op}={\mathfrak{F}}_{21}$. Also,
$u=m\big{(}(\mbox{id}\otimes s){\mathfrak{F}}\big{)}$ is an invertible element
in ${\mathfrak{A}}$ with
$u^{-1}=m\big{(}(s\otimes\mbox{id}){\mathfrak{F}}^{-1}\big{)}$, and the
twisted maps, $\tilde{\Delta}$ and $\tilde{s}$ are defined as

(3.2)
$\tilde{\Delta}(a)={\mathfrak{F}}\Delta(a){\mathfrak{F}}^{-1},\quad\tilde{s}=us(a)u^{-1}.$

######  Definition 3.4.

A twist ${\mathfrak{F}}$ is called admissible if it satisfies the co-cycle
condition:

(3.3) $({\mathfrak{F}}\otimes{\bf 1})\
(\Delta\otimes\mbox{id}){\mathfrak{F}}=({\bf 1}\otimes{\mathfrak{F}})\
(\mbox{id}\otimes\Delta){\mathfrak{F}}.$

In the special case that $\tilde{\cal R}={\bf 1}_{{\cal A}\otimes{\cal A}},$
then ${\cal R}=({\mathfrak{F}}^{(op)})^{-1}{\mathfrak{F}}$ and such a twist is
called factorizable.

We now introduce some convenient index notation:

(3.4)
${\mathfrak{F}}_{12,3}:=(\Delta\otimes\mbox{id}){\mathfrak{F}},\qquad{\mathfrak{F}}_{1,23}:=(\mbox{id}\otimes\Delta){\mathfrak{F}}.$

Then according to the co-cycle condition we introduce,
${\mathfrak{F}}_{123}={\mathfrak{F}}_{12}{\mathfrak{F}}_{12,3}={\mathfrak{F}}_{23}{\mathfrak{F}}_{1.23}$.
In general, we define

(3.5)
${\mathfrak{F}}_{12...n,n+1}:=(\Delta^{(n)}\otimes\mbox{id}){\mathfrak{F}},\qquad{\mathfrak{F}}_{1,2...n+1}:=(\mbox{id}\otimes\Delta^{(n)}){\mathfrak{F}}$

and consequently we define the $n$-twist, compatible also with the co-cycle
condition:

(3.6)
${\mathfrak{F}}_{12...n}={\mathfrak{F}}_{12...n-1}{\mathfrak{F}}_{12..n-1,n}={\mathfrak{F}}_{23..n}{\mathfrak{F}}_{1,2..n}.$

We introduce a similar notation for tensor representations of the quantum
algebra

(3.7) ${\mathfrak{T}}_{0,12..n}:=(\mbox{id}\otimes\Delta^{(n)}){\cal R}$

where recall $(\mbox{id}\otimes\Delta){\cal R}={\cal R}_{02}{\cal R}_{01}$ and
$(\Delta\otimes\mbox{id})={\cal R}_{10}{\cal R}_{20},$ thus we can explicitly
write ${\mathfrak{T}}_{0,12...n}={\cal R}_{0n}\ldots{\cal R}_{02}{\cal
R}_{01}$. The $n+1$ co-cycle condition and the explicit expressions of the
$n+1$ twist are also useful for our purposes here:

(3.8) $\displaystyle{\mathfrak{F}}_{012...n}$ $\displaystyle=$
$\displaystyle{\mathfrak{F}}_{01}{\mathfrak{F}}_{01,2}{\mathfrak{F}}_{012,3}\ldots{\mathfrak{F}}_{01...n-1,n}$
$\displaystyle=$
$\displaystyle{\mathfrak{F}}_{n-1n}{\mathfrak{F}}_{n-2,n-1n}\ldots{\mathfrak{F}}_{0,1..n-1n}.$

We state below three Propositions that include some of the main Drinfeld’s
results [11, 12] restricted to ${\cal R}$-matrices that satisfy the Yang-
Baxter equation. These results are most relevant to our present analysis that
follows in the next subsection. We also refer the interested reader to [24]
and references therein on similar proofs using the index notation.

######  Proposition 3.5.

(Drinfled 1) Let ${\cal R}$ be a solution of the Yang-Baxter equation and
${\mathfrak{F}}$ be an admissible twist such that, in the index notation,
$\tilde{\cal R}_{0n}={\mathfrak{F}}_{n0}\ {\cal R}_{0n}\
{\mathfrak{F}}^{-1}_{0n},$
where $n\in\\{1,\ldots N\\}$, then

(3.9) ${\mathfrak{F}}_{1...n-1n0n+1...N}{\cal R}_{0n}=\tilde{\cal
R}_{0n}{\mathfrak{F}}_{1...n-10nn+1...N}.$

###### Proof.

The proof relies on three basic statements:

  1. (1)

$\tilde{\cal R}_{0n}={\mathfrak{F}}_{n0}\ {\cal R}_{0n}\
{\mathfrak{F}}^{-1}_{0n}$

  2. (2)

${\mathfrak{F}}_{m-1,m...n0n+1...N}{\cal R}_{0n}={\cal
R}_{0n}{\mathfrak{F}}_{m-1,m...n-10n...N}$,
$\ m\leq n\in\\{1,\ldots N\\}$ and
${\mathfrak{F}}_{12...n0n+1...m,m+1}{\cal R}_{0n}={\cal
R}_{0n}{\mathfrak{F}}_{12...n-10n...m,m+1},$
$\ m\geq n\in\\{1,\ldots N\\}$

  3. (3)

the generalized co-cycle condition (3.8)

Statement (2) is a natural consequence of (3.5) and the fact that ${\cal R}$
satisfies the Yang-Baxter equation or equivalently conditions (1) and (2) of
Definition 3.3.

We start with

$\displaystyle{\mathfrak{F}}_{1...n-1n0n+1...N}{\cal R}_{0n}=$
$\displaystyle{\mathfrak{F}}_{1...n-1n0}{\mathfrak{F}}_{1...n-1n0,n+1}\ldots{\mathfrak{F}}_{1...n-1n0n+1...N-1,N}{\cal
R}_{0n}=$ (3.10) $\displaystyle{\mathfrak{F}}_{1...n-1n0}{\cal
R}_{0n}{\mathfrak{F}}_{1...n-10n,n+1}\ldots{\mathfrak{F}}_{1...n-10nn+1...N-1,N},$

where we have repeatedly used the generalized expressions (3.8) and statement
(2). We now focus on

(3.11) $\displaystyle{\mathfrak{F}}_{1...n-1n0}{\cal R}_{0n}$ $\displaystyle=$
$\displaystyle{\mathfrak{F}}_{n0}{\mathfrak{F}}_{n-1,n0}\ldots{\mathfrak{F}}_{1,2...n0}{\cal
R}_{0n}$ $\displaystyle=$ $\displaystyle{\mathfrak{F}}_{n0}{\cal
R}_{0n}{\mathfrak{F}}_{n-1,0n}\ldots{\mathfrak{F}}_{1,2...0n}$
$\displaystyle=$ $\displaystyle\tilde{\cal
R}_{0n}{\mathfrak{F}}_{0n}{\mathfrak{F}}_{n-1,0n}\ldots{\mathfrak{F}}_{1,2...0n}$
$\displaystyle=$ $\displaystyle\tilde{\cal R}_{0n}{\mathfrak{F}}_{12...0n}.$

Then by means of (3.11) statement (1) and (3.8) expression (3.10) becomes
$\tilde{\cal R}_{0n}{\mathfrak{F}}_{1...n-10nn+1...N},$ which concludes our
proof. ∎

We may now present the next two basic Propositions.

######  Proposition 3.6.

(Drinfeld 2) Let ${\cal R}$ satisfy the Yang-Baxter equation and $\tilde{\cal
R}={\mathfrak{F}}^{op}\ {\cal R}\ {\mathfrak{F}}^{-1}$, where ${\mathfrak{F}}$
is an admissible twist, then $\tilde{\cal R}$ also satisfies the YBE.

###### Proof.

We are employing the index notation together with Proposition 3.5. ${\cal R}$
satisfies the YBE, we multiply YBE with ${\mathfrak{F}}_{321}$ form the left
and ${\mathfrak{F}}_{123}^{-1}$ from the right:

(3.12) ${\mathfrak{F}}_{321}\ ({\cal R}_{12}\ {\cal R}_{13}\ {\cal
R}_{23}={\cal R}_{23}\ {\cal R}_{13}\ {\cal R}_{12})\
{\mathfrak{F}}_{123}^{-1}.$

We focus on the left side of the equation above, and use Proposition 3.5:

$\displaystyle{\mathfrak{F}}_{321}\ {\cal R}_{12}\ {\cal R}_{13}\ {\cal
R}_{23}\ {\mathfrak{F}}_{123}^{-1}=\tilde{\cal R}_{12}\ {\mathfrak{F}}_{312}\
{\cal R}_{13}\ {\cal R}_{23}\ {\mathfrak{F}}_{123}^{-1}=\tilde{\cal R}_{12}\
\tilde{\cal R}_{13}\ {\mathfrak{F}}_{132}\ {\cal R}_{23}\
{\mathfrak{F}}_{123}^{-1}=$ $\displaystyle\tilde{\cal R}_{12}\ \tilde{\cal
R}_{13}\ \tilde{\cal R}_{23}\ {\mathfrak{F}}_{123}\
{\mathfrak{F}}_{123}^{-1}=\tilde{\cal R}_{12}\ \tilde{\cal R}_{13}\
\tilde{\cal R}_{23}.$

Similarly, for the right hand side of (3.12) we end up with $\tilde{\cal
R}_{23}\ \tilde{\cal R}_{13}\ \tilde{\cal R}_{12}$, and this concludes our
proof. ∎

######  Proposition 3.7.

(Drinfeld 3) Let ${\cal R}$ be a solution of the Yang-Baxter equation and
${\mathfrak{F}}$ be an admissible twist: $\tilde{\cal R}={\mathfrak{F}}^{op}\
{\cal R}\ {\mathfrak{F}}^{-1}$. Let also ${\mathfrak{F}}_{012...N}$ defined in
(3.8), ${\mathfrak{T}}_{0,12..N}:={\cal R}_{0N}\ldots{\cal R}_{01}$ and
$\tilde{\mathfrak{T}}_{0,12...N}:=\tilde{\cal R}_{0N}\ldots\tilde{\cal
R}_{01}$, then

(3.13) ${\mathfrak{F}}_{12...N0}\ {\mathfrak{T}}_{0,12..N\ }\
{\mathfrak{F}}^{-1}_{012..N}=\tilde{\mathfrak{T}}_{0,12...N}$

###### Proof.

The proof of this Proposition follows exactly the same logic as the proof of
Proposition 3.6. We start with the LHS of (3.13), and use Proposition 3.5:

$\displaystyle{\mathfrak{F}}_{12...N0}\ {\cal R}_{0N}\ {\cal
R}_{0N-1}\ldots{\cal R}_{01}\ {\mathfrak{F}}^{-1}_{012..N}=$
$\displaystyle\tilde{\cal R}_{0N}\ {\mathfrak{F}}_{12...N-10N}\ {\cal
R}_{0N-1}\ldots{\cal R}_{01}\ {\mathfrak{F}}^{-1}_{012..N}=\ldots$
$\displaystyle\tilde{\cal R}_{0N}\ \tilde{\cal R}_{0N-1}\ldots\tilde{\cal
R}_{01}\ {\mathfrak{F}}_{012...N-1N}\
{\mathfrak{F}}^{-1}_{012..N}\equiv\tilde{\mathfrak{T}}_{0.12...N}.$

∎

###  3.2. Quasi-admissible twists for set-theoretic solutions

We are focusing henceforth on $R$ matrices acting of tensor products of finite
${\cal N}$ dimensional spaces $V,$ i.e. $R:V\otimes V\to V\otimes V.$ Our main
aim is to identify the explicit form of Drinfeld’s twists for finite,
involutive, non-degenerate, set-theoretic solutions of the YBE coming from
braces, and also show that they are admissible. We are employing the term
quasi-admissible twist due to the fact that we only show here the
admissibility of the set-theoretic twists, i.e. the validity of the co-cycle
condition, however we are not exploring the action of a co-unit on such
twists. We make some preliminary remarks at the end of the manuscript
illustrating the singular nature of set-theoretic twists via a simple example.

More specifically, in the analysis that follows we focus on finite,
involutive, non-degenerate, set-theoretic solutions of the YBE given by
$r={\cal P}\check{r}$, where
$\check{r}=\sum_{x,y,\in X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}$. We
review below the fundamental constraints emerging from the YBE for such
solutions.

Constraints for set-theoretic solutions of the YBE. For any finite, non-
degenerate, involutive, set-theoretic solution of the YBE $r={\cal
P}\check{r}$, where
$\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}$, the
following conditions hold:

(3.14)
$\sigma_{\sigma_{\eta}(\hat{x})}(\sigma_{\tau_{x}(\eta)}(y))=\sigma_{\eta}(\sigma_{x}({y}))\quad\mbox{and}\quad\tau_{\tau_{y}(x)}(\tau_{\sigma_{x}(y)}(\eta))=\tau_{y}(\tau_{x}(\eta)),$

$\forall\ \eta,x,y\in X$: $\sigma_{\eta}(x),\ \tau_{y}(x)$ are fixed.

###### Proof.

It is convenient for our purposes here to express the Yang-Baxter equation in
the following form:

(3.15) $\check{r}_{23}\ T_{1,23}=T_{1,23}\ \check{r}_{23}$

(similarly $\check{r}_{12}\ T_{12,3}=T_{12,3}\ \check{r}_{12}$), where we
define, $T_{1,23}=r_{13}\ r_{12}$, ($T_{12,3}=r_{13}\ r_{23}$) and recall
$\check{r}={\cal P}r.$

We take the following steps.

  1. (1)

We first identify $T_{1,23}=r_{13}r_{12}$ , where $r={\cal P}\check{r}$ and
$\check{r}$ is the set-theoretic solution

(3.16) $T_{1,23}=\sum_{x,y,\eta\in X}e_{\tau_{y}(x),\sigma_{\eta}(x)}\otimes
e_{\eta,\tau_{x}(\eta)}\otimes e_{\sigma_{x}(y),y}.$

  2. (2)

We calculate the left and right hand side of (3.15), using also the involutive
property,

(3.17) $\displaystyle LHS$ $\displaystyle=$ $\displaystyle\sum_{x,y,\eta\in
X}e_{\tau_{y}(x),\sigma_{\eta}(x)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),\tau_{x}(\eta)}\otimes
e_{\tau_{\sigma_{x}(y)}(\eta),y}$ (3.18) $\displaystyle RHS$ $\displaystyle=$
$\displaystyle\sum_{x,y,\eta\in X}e_{\tau_{y}(x),\sigma_{\eta}(x)}\otimes
e_{\eta,\sigma_{\tau_{x}(\eta)}(y)}\otimes
e_{\sigma_{x}(y),\tau_{y}(\tau_{x}(\eta))}.$

  3. (3)

We identify the constraints emerging from (3.15) by equating expressions
(3.17) and (3.18). Equivalence of expression (3.17) and (3.18) leads to

(3.19)
$\displaystyle\eta=\sigma_{\hat{\eta}}(\sigma_{\hat{x}}(\hat{y})),\quad\sigma_{x}(y)=\tau_{\sigma_{\hat{x}}(\hat{y})}(\hat{\eta})\quad\mbox{and}$
(3.20)
$\displaystyle\tau_{x}(\eta)=\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y}),\quad
y=\tau_{\hat{y}}(\tau_{\hat{x}}(\hat{\eta})).$

Note that relations (3.19) are equivalent to:
$\sigma_{\eta}(\sigma_{x}(y))=\hat{\eta},$ and
$\tau_{\sigma_{x}(y)}(\eta)=\sigma_{\hat{x}}(\hat{y})$. We also require

(3.21)
$\sigma_{\eta}(x)=\sigma_{\hat{\eta}}(\hat{x})\quad\mbox{and}\quad\tau_{y}(x)=\tau_{\hat{y}}(\hat{x}).$

Conditions (3.19)-(3.21) guarantee the equivalence between the LHS and RHS of
(3.15) given in (3.17), (3.18).

  4. (4)

We consider together (3.20), (3.21) and obtain
$\sigma_{\eta}(x)=\sigma_{\hat{\eta}}(\hat{x}),\
\tau_{x}(\eta)=\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y})$, which leads to
$\eta=\sigma_{\sigma_{\hat{\eta}}(\hat{x})}(\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y}))$,
then comparing with (3.19) we obtain the first fundamental constraint:

(3.22)
$C_{1}\equiv\sigma_{\sigma_{\hat{\eta}}(\hat{x})}(\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y}))-\sigma_{\hat{\eta}}(\sigma_{\hat{x}}({\hat{y}}))=0,$

and conversely if $C_{1}=0$ then
$\sigma_{\eta}(x)=\sigma_{\hat{\eta}}(\hat{x})$ given (3.19), (3.20).

Similarly, from (3.19), (3.21)
$\sigma_{x}(y)=\tau_{\sigma_{\hat{x}}(\hat{y})}(\hat{\eta})$,
$\tau_{y}(x)=\tau_{\hat{y}}(\hat{x})$, which leads to
$y=\tau_{\tau_{\hat{y}}(\hat{x})}(\tau_{\sigma_{\hat{x}}(\hat{y})}(\hat{\eta}))$,
and comparison with (3.20) leads to the second fundamental constraint:

(3.23)
$C_{2}\equiv\tau_{\tau_{\hat{y}}(\hat{x})}(\tau_{\sigma_{\hat{x}}(\hat{y})}(\hat{\eta}))-\tau_{\hat{y}}(\tau_{\hat{x}}(\hat{\eta}))=0,$

and conversely if $C_{2}=0$ then $\tau_{\hat{y}}(\hat{x})=\tau_{y}(x),$ given
(3.19), (3.20).

For the brace solution of the YBE, which is our main interest in this study,
the above constraints are satisfied as shown by Rump. ∎

We derive in the following Lemma the explicit forms of the co-products of the
quantum algebra associated to brace solutions of the YBE. This will be of
significance when identifying the corresponding admissible Drinfeld twists.

######  Lemma 3.8.

Let $\check{r}$ be an involutive, finite, non-degenerate set-theoretic
solution of the YBE and $T_{1,23}=\sum_{x,y\in X}e_{x,y}\otimes\Delta({\cal
L}_{x,y})$, where ${\cal L}_{x,y}$ are the represented elements of the
corresponding quantum algebra defined by the RTT relation (2.7) and $r={\cal
P}\check{r}$ $({\cal P}$ is the permutation operator). Then

(3.24) $\Delta({\cal L}_{x,y})\ \check{r}=\check{r}\ \Delta({\cal
L}_{x,y})\quad\forall x,\ y\in X,$

subject to (3.22), (3.23).

###### Proof.

The proof is an immediate consequence of the constraints emerging for set-
theoretic solutions of the YBE. Indeed, from (3.16) we can read of

(3.25) $\Delta({\cal L}_{\tau_{y}(x),\sigma_{\eta}(x)})=\sum_{x,y,\eta\in
W}e_{\eta,\tau_{x}(\eta)}\otimes e_{\sigma_{x}(y),y},$

where $W:x,y,\eta\in X$ such that $\sigma_{\eta}(x)$ and $\tau_{y}(x)$ are
fixed.
Then due to $\check{r}_{23}\ T_{1,23}\ =T_{1,23}\ \check{r}_{23}$ we obtain
(3.24). i.e. the entries of the $T_{1,23}$-matrix commute with $\check{r}$,
subject to (3.22), (3.23). ∎

It was shown in [8] Proposition 3.3 that any brace solution of the YBE can be
obtained from the permutation operator via a similarity transformation, i.e.
$\check{r}_{12}={\cal F}^{-1}_{12}{\cal P}_{12}{\cal F}_{12}$ 222The twist of
the present study is the inverse of ${\cal F}$ in [8]. It is just a matter of
convention.

######  Proposition 3.9.

Let $\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}$ be
the brace solution of the braid YBE. Let also $V_{k}$, $k\in\\{1,\ldots,{\cal
N}^{2}\\}$ be the eigenvectors of the permutation operator ${\cal
P}=\sum_{x,y\in X}e_{x,y}\otimes e_{y,x}$, and $\hat{V}_{k}$,
$k\in\\{1,\ldots,{\cal N}^{2}\\}$ be the eigenvectors of the brace
$\check{r}$-matrix. Then the $\check{r}$-matrix can be expressed as a Drinfeld
twist, such that $\check{r}={\cal F}^{-1}{\cal P}{\cal F}$, where the twist
${\cal F}^{-1}$ is expressed as ${\cal F}^{-1}=\sum_{k=1}^{{\cal
N}^{2}}\hat{V}_{k}\ V_{k}^{T}$ [8].

From the proof of the above Proposition (Proposition 3.3 in [8]) we know that
the eigenvectors of the permutation operator ${\cal P}$ and the set-theoretic
braid solution $\check{r}$ are given as follows. Let $\hat{e}_{k},\
k\in\\{1,\ldots,{\cal N}\\}$ be the ${\cal N}$ dimensional column vector with
1 in the $k^{th}$ position and 0 elsewhere, i.e. $\hat{e}_{k}$ form a basis of
the ${\cal N}$ dimensional vector space.

  1. (1)

The (normalized) eigenvectors of the permutation operator are ($x,\ y\in X$):

$\displaystyle V_{k}=\hat{e}_{x}\otimes\hat{e}_{x},\quad
k\in\big{\\{}1,\ldots,{\cal N}\big{\\}}$ $\displaystyle
V_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{y}+\hat{e}_{y}\otimes\hat{e}_{x}\big{)},\quad
x\neq y,\quad k\in\big{\\{}{\cal N}+1,\ldots,{{\cal N}^{2}+{\cal N}\over
2}\big{\\}},$ (3.26) $\displaystyle
V_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{y}-\hat{e}_{y}\otimes\hat{e}_{x}\big{)},\quad
x\neq y,\quad k\in\big{\\{}{{\cal N}^{2}+{\cal N}\over 2}+1,\ldots,{\cal
N}^{2}\big{\\}}.$

The first ${{\cal N}^{2}+{\cal N}\over 2}$ eigenvectors have the same
eigenvalue $1$, while the rest ${{\cal N}^{2}-{\cal N}\over 2}$ eigenvectors
have eigenvalue $-1$. Also it is easy to check that $V_{k}$ form an ortho-
normal basis for the ${\cal N}^{2}$ dimensional space. Indeed,
$V^{T}_{k}V_{l}=\delta_{kl}$ and $\sum_{k=1}^{{\cal
N}^{2}}V_{k}V_{k}^{T}=I_{{\cal N}^{2}}$ (T denotes usual transposition).

  2. (2)

The eigenvectors of the $\check{r}$-matrix are

$\displaystyle\hat{V}_{k}=\hat{e}_{x}\otimes\hat{e}_{y},\quad(x,y)=(\sigma_{x}(y),\tau_{y}(x)),\quad
k\in\big{\\{}1,\ldots,{\cal N}\big{\\}}$
$\displaystyle\hat{V}_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{y}+\hat{e}_{\sigma_{x}(y)}\otimes\hat{e}_{\tau_{y}(x)}\big{)},~{}~{}~{}~{}~{}~{}k\in\big{\\{}{\cal
N}+1,\ldots,{{\cal N}^{2}+{\cal N}\over 2}\big{\\}},$
$\displaystyle\hat{V}_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{y}-\hat{e}_{\sigma_{x}(y)}\otimes\hat{e}_{\tau_{y}(x)}\big{)},~{}~{}~{}~{}(x,y)\neq(\sigma_{x}(y),\tau_{y}(x)),$
(3.27) $\displaystyle k\in\big{\\{}{{\cal N}^{2}+{\cal N}\over
2}+1,\ldots,{\cal N}^{2}\big{\\}}$

As in the case of the permutation operator the $\check{r}$ matrix has the same
eigenvalues $1$ and $-1$ and the same multiplicities, ${{\cal N}^{2}+{\cal
N}\over 2}$ and ${{\cal N}^{2}-{\cal N}\over 2}$ respectively. Hence, the two
matrices are similar, i.e. there exists some invertible ${\cal
F}\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes\mbox{End}({\mathbb{C}}^{\cal
N})$ (not uniquely defined) such that $\check{r}={\cal F}^{-1}{\cal P}{\cal
F}$, where ${\cal F}^{-1}=\sum_{k=1}^{{\cal N}^{2}}\hat{V}_{k}\ V_{k}^{T}.$

From Proposition 3.3 in [8] we can extract explicit forms for the twist ${\cal
F}$ and state the following.

######  Proposition 3.10.

Let ${\cal F}^{-1}=\sum_{k=1}^{{\cal N}^{2}}\hat{V}_{k}\ V_{k}^{T}$ be the
similarity transformation (twist) of Proposition 3.9, such that
$\check{r}={\cal F}^{-1}{\cal P}{\cal F},$ where $\check{r}=\sum_{x,y\in
X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}$ is the brace solution of the
braid YBE, ${\cal P}$ is the permutation operator and $\hat{V}_{k},\ V_{k}$
are their respective eigenvectors. Then the twist can be explicitly expressed
as ${\cal F}=\sum_{x,y\in X}e_{x,x}\otimes e_{\sigma_{x}(y),y}$.

###### Proof.

We begin our proof by re-expressing the eigenvectors of the permutation
operator in a convenient for our purposes form. The first ${\cal N}$
eigenvectors are used as they are, we only conveniently re-express the rest
(let $n={{\cal N}^{2}+{\cal N}\over 2}$):

$\displaystyle V_{k}=\hat{e}_{x}\otimes\hat{e}_{x},\quad
k\in\big{\\{}1,\ldots,{\cal N}\big{\\}}$ $\displaystyle
V_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{\sigma_{x}(y)}+\hat{e}_{\sigma_{x}(y)}\otimes\hat{e}_{x}\big{)},~{}~{}x\neq\sigma_{x}(y),~{}~{}~{}k\in\big{\\{}{\cal
N}+1,\ldots,n\big{\\}},$ (3.28) $\displaystyle
V_{k}={1\over\sqrt{2}}\big{(}\hat{e}_{x}\otimes\hat{e}_{\sigma_{x}(y)}-\hat{e}_{\sigma_{x}(y)}\otimes\hat{e}_{x}\big{)},~{}~{}x\neq\sigma_{x}(y),~{}~{}k\in\big{\\{}n+1,\ldots,{\cal
N}^{2}\big{\\}}.$

Let us first mention that if $x=\sigma_{x}(y)$ then $y=\tau_{x}(y)$ (and vice
versa). Indeed, this can be show as follows: let $\hat{y}=\tau_{y}(x),$ then
due to involution we obtain $x=\sigma_{x}(\hat{y})$ (and
$y=\tau_{\hat{y}}(x)$), but due to the fact that $\sigma_{x}(y)$ is a
bijection, and also $x=\sigma_{x}(y),$ we conclude that $\hat{y}=y$. We may
now compute ${\cal F}^{-1}$ using the explicit expressions of the eigenvectors
of $\check{r}$ and ${\cal P}$ (3.27), (3.28) (recall $n={{\cal N}^{2}+{\cal
N}\over 2}$) $x,\ y\in X$:

$\displaystyle{\cal F}^{-1}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{{\cal
N}}\hat{V}_{k}\ V_{k}^{T}+\sum_{k={\cal N}+1}^{n}\hat{V}_{k}\
V_{k}^{T}+\sum_{k=n+1}^{{\cal N}^{2}}\hat{V}_{k}\ V_{k}^{T}$ $\displaystyle=$
$\displaystyle\sum_{x=\sigma_{x}(y)}(\hat{e}_{x}\otimes\hat{e}_{y})(\hat{e}_{x}^{T}\otimes\hat{e}_{x}^{T})$
$\displaystyle+$ $\displaystyle{1\over
2}\sum_{x\neq\sigma_{x}(y)}(\hat{e}_{x}\otimes\hat{e}_{y})(\hat{e}_{x}^{T}\otimes\hat{e}_{\sigma_{x}(y)}^{T})+{1\over
2}\sum_{x\neq\sigma_{x}(y)}(\hat{e}_{\sigma_{x}(y)}\otimes\hat{e}_{\tau_{y}(x)})(\hat{e}_{\sigma_{x}(y)}^{T}\otimes\hat{e}_{x}^{T})$
$\displaystyle=$ $\displaystyle\sum_{x=\sigma_{x}(y)}e_{x,x}\otimes
e_{y,x}+\sum_{x\neq\sigma_{x}(y)}e_{x,x}\otimes e_{y,\sigma_{x}(y)}$
$\displaystyle=$ $\displaystyle\sum_{x,y\in X}e_{x,x}\otimes
e_{y,\sigma_{x}(y)}$

From the above expression and due to the fact that $\sigma_{x}$ and $\tau_{y}$
are bijections, we conclude that ${\cal F}=\sum_{x,y\in X}e_{x,x}\otimes
e_{\sigma_{x}(y),y}$, (${\cal F}_{12}^{-1}={\cal F}_{12}^{T},$ where T denotes
total transposition in both spaces).

Recall that $r={\cal P}\check{r},$ we also confirm by direct computation, and
using the fact that $\sigma_{x},\ \tau_{y}$ are bijections that $({\cal
F}^{(op)})^{-1}{\cal F}=\sum_{x,\in X}e_{y,\sigma_{x}(y)}\otimes
e_{x,\tau_{y}(x)}=r$. ∎

######  Remark 3.11.

The twist is not uniquely defined, for instance an alternative twist is of the
form ${\cal G}=\sum_{x,y\in X}e_{\tau_{y}(x),x}\otimes e_{y,y}$, and
$\sum_{x,\in X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}={\cal
G}_{21}^{-1}{\cal G}_{12}$. This is shown by direct computation.

######  Remark 3.12.

The Baxterized solution of the YBE, $R(\lambda)=\lambda r+{\cal P}$ (recall
${\cal P}$ is the permutation operator) can be also expressed as
$R_{12}(\lambda)={\cal F}^{-1}_{21}\tilde{R}_{12}(\lambda){\cal F}_{12}$,
where $\tilde{R}$ is the Yangian. This is a straightforward statement due to
the form of the Yangian $\tilde{R}(\lambda)=\lambda{\mathbb{I}}+{\cal P}$, and
due to the fact that ${\cal F}_{21}^{-1}{\cal P}_{12}{\cal F}_{12}={\cal
P}_{12}$.

We now focus on the derivation of the 3-twist ${\cal F}_{012}$ and the co-
cycle condition. We will now identify the 3-twist for any finite, involutive,
non-degenerate set-theoretic solution and inspired by Propositions (3.5)-(3.7)
will show that a suitable co-cycle condition is satisfied, i.e. ${\cal F}$ is
an admissible twist. This is achieved in the following Propositions.

######  Proposition 3.13.

In accordance to the co-multiplication of the quantum algebra as defined in
Lemma 3.8, we define:

(3.29) $\displaystyle{\cal F}_{0,12}=\sum_{x,y,\eta\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes e_{\eta,\tau_{x}(\eta)}\
\otimes e_{\sigma_{x}(y),y}|_{C_{1}=0}$ (3.30) $\displaystyle{\cal
F}_{01,2}=\sum_{x,y,\eta\in X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes
e_{\tau_{x}(\eta),\tau_{x}(\eta)}\ \otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}|_{C_{1}=0}$

where
$C_{1}=\sigma_{\sigma_{\eta}(x)}(\sigma_{\tau_{x}(y)}(y))-\sigma_{\eta}(\sigma_{x}(y))$.
Let also $\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes
e_{y,\tau_{y}(x)}$, then

(3.31) $\check{r}_{01}{\cal F}_{01,2}={\cal F}_{01,2}\check{r}_{01},\
\quad\check{r}_{12}{\cal F}_{0,12}={\cal F}_{0,12}\check{r}_{12}.$

###### Proof.

We divide the proof in two parts.

  1. (1)

We first show that $\check{r}_{01}{\cal F}_{01,2}={\cal
F}_{01,2}\check{r}_{01}$, using the explicit expressions of $\check{r}$,
${\cal F}_{0,12}$ and ${\cal F}_{01,2}$: The LHS of the expression is (subject
to the constraint $C_{1}=0$)

$\displaystyle\check{r}_{01}{\cal F}_{01,2}=\sum_{\eta,x,y\in
X}e_{\eta,\sigma_{\eta}(x)}\otimes e_{x,\tau_{x}(\eta)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}.$

Similarly,

$\displaystyle{\cal F}_{01,2}\check{r}_{01}=\sum_{\eta,x,y\in
X}e_{\eta,\sigma_{\eta}(x)}\otimes e_{x,\tau_{x}(\eta)}\otimes
e_{\sigma_{\sigma_{\eta}(x)}(\sigma_{\tau_{x}(\eta)}(y)),y},$

and due to the constraint $C_{1}=0$: ${\cal
F}_{01,2}\check{r}_{01}=\sum_{\eta,x,y\in X}e_{\eta,\sigma_{\eta}(x)}\otimes
e_{x,\tau_{x}(\eta)}\otimes e_{\sigma_{\eta}(\sigma_{x}(y)),y}$, which
concludes the first part of our proof.

  2. (2)

We move on now to the second part of our proof, which is a bit more involved:
$\check{r}_{12}{\cal F}_{0,12}={\cal F}_{0,12}\check{r}_{12}$. By direct
computation given the explicit expressions for ${\cal F}_{0,12}$ and
$\check{r}_{12}$:

$\displaystyle\check{r}_{12}{\cal F}_{0,12}=\sum_{\eta,x,y\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),\tau_{x}(\eta)}\otimes
e_{\tau_{\sigma_{x}(y)}(\eta),y}$ $\displaystyle{\cal
F}_{0,12}\check{r}_{12}=\sum_{\eta,x,y\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes
e_{h,\sigma_{\tau_{x}(\eta)}(y)}\otimes
e_{\sigma_{x}(y),\tau_{y}(\tau_{x}(\eta))},$

subject to $C_{1}=0$. We now follow the logic of obtaining the constraints for
set-theoretic solutions of the YBE. We employ (3.19) and (3.20) which as
argued due to $C_{1}=0$ guarantee that
$\sigma_{\eta}(x)=\sigma_{\hat{\eta}}(\hat{x})$ leading to
$\check{r}_{12}{\cal F}_{0,12}={\cal F}_{0,12}\check{r}_{12}$, which basically
concludes our proof.

All possible permutations among the indices $0,1,2$ can be considered in a
straightforward manner. For instance, by multiplying expression
$\check{r}_{12}{\cal F}_{0,12}={\cal F}_{0,12}\check{r}_{12}$ with ${\cal
P}_{01}$ (left and right) we obtain $\check{r}_{02}{\cal F}_{1,02}={\cal
F}_{1,02}\check{r}_{02}$. ∎

######  Remark 3.14.

This is a straightforward, but useful remark. Recall, $r={\cal P}\check{r}$,
where ${\cal P}$ is the permutation operator. If ${\cal
F}_{01,2}\check{r}_{01}=\check{r}_{01}{\cal F}_{01,2},$ and ${\cal
F}_{1,02}\check{r}_{02}=\check{r}_{02}{\cal F}_{1,02},$ as in Proposition
3.13, then by multiplying the latter two equalities with ${\cal P}$ from the
left we conclude: ${\cal F}_{10,2}r_{01}=r_{01}{\cal F}_{01,2},$ and ${\cal
F}_{1,20}r_{02}=r_{02}{\cal F}_{1,02}.$

Generalization. The $(n+1)$-objects ${\cal F}_{0,12...n}$ and ${\cal
F}_{0,12...n}$ can be now derived by iteration exploiting the explicit form of
$T_{0,12..n}=r_{0n}\ldots r_{01}$. Indeed, let us first identify $T_{0,1..n}$
recalling the specific form of the set-theoretic solution $r=\sum_{x,y\in
X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)},$

(3.32) $T_{0,1..n}=\sum_{x_{1},..x_{n},y_{1}\in
X}e_{y_{n},\sigma_{x_{1}}(y_{1})}\otimes
e_{x_{1},\tau_{y_{1}}(x_{1})}\otimes\ldots\otimes
e_{x_{n},\tau_{y_{n}}(x_{n})},$

subject to $~{}y_{m-1}=\sigma_{x_{m}}(y_{m}),$ $m\in\\{2,\ldots,n\\}$. Then
according to the generic expression (3.32), (bear in mind that
$T_{0,1..n}\check{r}_{mm+1}=\check{r}_{mm+1}T_{0,1..n},$
$m\in\\{1,\ldots,n-1\\};$ see also Proposition 3.13 and expression (3.29)) we
derive,
${\cal F}_{0,1...n}=\sum_{x_{1},...,x_{n},y_{n}\in
X}e_{\sigma_{x_{1}}(y_{1}),\sigma_{x_{1}}(y_{1})}\otimes
e_{x_{1},\tau_{y_{1}}(x_{1})}\otimes\ldots\otimes
e_{x_{n},\tau_{y_{n}}(x_{n})},$ subject to $~{}y_{m-1}=\sigma_{x_{m}}(y_{m}),$
$m\in\\{2,\ldots,n\\}$. Also, by iteration (see expression (3.30)) we obtain,
${\cal F}_{01..n-1,n}=\sum_{x_{1},...,x_{n+1}\in X}e_{x_{1},x_{1}}\otimes
e_{x_{2},x_{2}}\otimes\ldots e_{x_{n},x_{n}}\otimes e_{{\mathrm{X}},x_{n+1}},$
where we define
${\mathrm{X}}:=\sigma_{x_{1}}(\sigma_{x_{2}}(...(\sigma_{x_{n}}(x_{n+1}))...)$.
The general expressions hold subject to the constraint
$\sigma_{x_{n}}(\sigma_{x_{n+1}}(y))=\sigma_{\sigma_{x_{n}}(x_{n+1})}(\sigma_{\tau_{x_{n+1}}(x_{n})}(y)),$
$\ x_{n},y\in X$.

We may now proceed in proving the admissibility of the derived twist, i.e.
show the validity of the co-cycle condition.

######  Proposition 3.15.

Let ${\cal F}_{01}={\cal F}\otimes I$ and ${\cal F}_{12}=I\otimes{\cal F}$,
where ${\cal F}=\sum_{\eta,x,y\in X}e_{\eta,\eta}\otimes
e_{\sigma_{\eta}(x),x}$. Let also ${\cal F}_{01,2}$ and ${\cal F}_{0,12}$
defined in (3.29) and (3.30). Then

(3.33) ${\cal F}_{012}:={\cal F}_{01}{\cal F}_{01,2}={\cal F}_{12}{\cal
F}_{0,12}.$

###### Proof.

By substituting the expressions for ${\cal F}_{01},\ {\cal F}_{12}$, ${\cal
F}_{01,2}$ and ${\cal F}_{0,12}$ (recall $C_{1}=0$ holds for ${\cal
F}_{01,2},$ ${\cal F}_{0,12}$) we obtain by direct computation:

$\displaystyle{\cal F}_{01}{\cal F}_{01,2}=\sum_{\eta,x,y\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes e_{\eta,\tau_{x}(\eta)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}.$

Similarly,

$\displaystyle{\cal F}_{12}{\cal F}_{0,12}=\sum_{\eta,x,y\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes e_{\eta,\tau_{x}(\eta)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}$

The explicit form the 3-twist is given from the expressions above as
${\cal F}_{012}=\sum_{\eta,x,y\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes e_{\eta,\tau_{x}(\eta)}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}|_{C_{1}=0}.$ ∎

We are now in the position to show the factorization of the monodromy matrix
$T_{0,12}$ in terms of the admissible twists.

######  Corollary 3.16.

Let $r=\sum_{x,y\in X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}$, a
solution of the YBE, and ${\cal F}_{012}=\sum_{x,y,\eta\in
X}e_{\eta,\eta}\otimes e_{\sigma_{\eta}(x),x}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}|_{C_{1}=0},$ as defined in Proposition
3.15. Let also, $T_{0,12}=r_{02}r_{01}$, then $T_{0,12}={\cal F}^{-1}_{120}\
{\cal F}_{012}$ .

###### Proof.

This is based on Proposition 3.7. Indeed, to show the decomposition of the
monodromy matrix $T_{0,12}$ constructed from set-theoretic solutions we employ
the following statements:

  1. (1)

Proposition 3.7: ${\cal F}_{n0}^{-1}\ {\cal F}_{0n}=r_{0n}$, $n\in\\{1,\
2\\}$.

  2. (2)

Proposition 3.13 and Remark 3.14: ${\cal F}_{n0,m}r_{0n}=r_{0n}{\cal
F}_{0n,m}$ and
${\cal F}_{m,n0}r_{0n}=r_{0n}{\cal F}_{m,0n},$ $n\neq m\in\\{1,\ 2\\}$.

  3. (3)

Proposition 3.15: the co-cycle condition, ${\cal F}_{01}{\cal F}_{01,2}={\cal
F}_{12}{\cal F}_{0,12}$.

We first recall that ${\cal F}_{n0,m}={\cal P}_{0n}{\cal F}_{0n,m}{\cal
P}_{0n},$ similarly for ${\cal F}_{m,n0}$, where ${\cal P}$ is the permutation
operator. The proof is straightforward based on the proof of Proposition 3.7:
i.e. use (1)-(3) above, then ${\cal F}_{120}\ T_{0,12}\ {\cal
F}_{012}^{-1}=I_{V^{\otimes 3}}$. ∎

Direct computation. Let us also confirm Corollary 3.16 by direct computation.
We first derive $T_{0,12}$ via (3.32) for $n=2$:

(3.34) $\displaystyle T_{0,12}=\sum_{\eta,x,y\in
X}e_{y,\sigma_{\eta}(\sigma_{x}(y))}\otimes
e_{\eta,\tau_{\sigma_{x}(y)}(\eta)}\otimes e_{x,\tau_{y}(x)}.$

We also derive ${\cal F}_{012}^{-1}$; due to the fact that $\sigma_{y},\
\tau_{x}$ are bijective functions: ${\cal F}_{012}^{-1}={\cal F}_{012}^{T}$,
where T denotes total transposition, i.e. transposition in all three spaces of
the tensor product. We may now show by direct computation that $T_{0,12}={\cal
F}_{120}^{-1}\ {\cal F}_{012}$, indeed

$\displaystyle{\cal F}_{120}^{-1}\ {\cal F}_{012}=$
$\displaystyle\Big{(}\sum_{\eta,x,y\in
X}e_{y,\sigma_{\eta}(\sigma_{x}(y))}\otimes e_{\eta,\eta}\otimes
e_{x,\sigma_{\eta}(x)}\Big{)}\Big{(}\sum_{\hat{\eta},\hat{x},\hat{y}\in
X}e_{\hat{\eta},\hat{\eta}}\otimes
e_{\sigma_{\hat{\eta}}(\hat{x}),\hat{x}}\otimes
e_{\sigma_{\hat{\eta}}(\sigma_{\hat{x}}(\hat{y})),\hat{y}}\Big{)}=$ (3.35)
$\displaystyle\sum_{x,y,\eta,\hat{x},\hat{y},\hat{\eta}\in
X}e_{y,\hat{\eta}}\otimes e_{\eta,\hat{x}}\otimes e_{x,\hat{y}}$

subject to the following conditions:

(3.36)
$\hat{\eta}=\sigma_{\eta}(\sigma_{x}(y)),\quad\sigma_{\hat{\eta}}(\hat{x})=\eta,\quad\sigma_{\eta}(x)=\sigma_{\hat{\eta}}(\sigma_{\hat{x}}(\hat{y}))$

Our aim now is to express the $\hat{x},\hat{y},\hat{\eta}$ in terms of
$x,y,\eta$, notice that $\hat{\eta}$ is already expressed in such a way.
Consider now the condition $\sigma_{\hat{\eta}}(\hat{x})=h$, and let
$\tau_{\hat{x}}(\hat{\eta})=\xi$ then we obtain via the involutive property:
$\hat{\eta}=\sigma_{\eta}(\xi)$ and $\hat{x}=\tau_{\xi}(\eta)$. However, due
to the first of the conditions (3.36) and the fact that $\sigma_{\eta}$ is a
bijective function, we conclude that $\xi=\sigma_{x}(y)$ and hence
$\hat{x}=\tau_{\sigma_{x}(y)}(\eta)$. It remains now to express $\hat{y}$ in
terms of $x,y,\eta$; we consider the third of the conditions (3.36) as well as
condition $C_{1}=0$ then
$\sigma_{\eta}(x)=\sigma_{\sigma_{\hat{\eta}}(\hat{x})}(\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y}))$,
but as shown above $\sigma_{\hat{\eta}}(\hat{x})=h$, then using also the fact
that $\sigma_{\eta}$ is a bijection we conclude
$x=\sigma_{\tau_{\hat{x}}(\hat{\eta})}(\hat{y})$. From our considerations
above $\tau_{\hat{x}}(\hat{\eta})(=\xi)=\sigma_{x}(y),$ and we conclude that
$\hat{y}=\tau_{y}(x)$.

Having expressed $\hat{x},\hat{y},\hat{\eta}$ in terms of $x,y,\eta$:
$\hat{\eta}=\sigma_{\eta}(\sigma_{x}(y)),$
$\hat{x}=\tau_{\sigma_{x}(y)}(\eta),$ and $\hat{y}=\tau_{y}(x),$ we arrive via
(3.35) at

(3.37) ${\cal F}_{120}^{-1}\ {\cal F}_{012}=\sum_{x,y,\eta\in
X}e_{y,\sigma_{\eta}(\sigma_{x}(y))}\otimes
e_{\eta,\tau_{\sigma_{x}(y)}(\eta)}\otimes e_{x,\tau_{y}(x)}$

which is precisely $T_{0,12}$ (3.34). $\hskip 224.03743pt\square$

######  Remark 3.17.

An alternative admissible twist is derived as follows. Using the notation
introduced in (3.5), we define

(3.38) $\displaystyle{\cal G}_{0,12}=\sum_{\eta,x,y\in
X}e_{\tau_{y}(\tau_{x}(\eta)),\eta}\otimes
e_{\sigma_{x}(y),\sigma_{x}(y)}\otimes e_{\tau_{y}(x),\tau_{y}(x)}|_{C_{2}=0}$
(3.39) $\displaystyle{\cal G}_{01,2}=\sum_{\eta,x,y\in
X}e_{\tau_{x}(\eta),\eta}\otimes e_{y,\sigma_{x}(y)}\otimes
e_{\tau_{y}(x),\tau_{y}(x)}|_{C_{2}=0}$

$C_{2}=\tau_{y}(\tau_{x}(\eta))-\tau_{\tau_{y}(x)}(\tau_{\sigma_{x}(y)}(\eta))$,
then:

  1. (1)

${\cal G}_{01,2}\check{r}_{01}=\check{r}_{01}{\cal G}_{01,2}$ and ${\cal
G}_{0,12}\check{r}_{12}=\check{r}_{12}{\cal G}_{0,12}$, where $\check{r}$ is
the brace solution.

  2. (2)

The co-cycle condition is also satisfied and the 3-twist is then derived, i.e.
${\cal G}_{012}:={\cal G}_{01}{\cal G}_{01,2}={\cal G}_{12}{\cal G}_{0,12}$,
where the 2-twist is ${\cal G}=\sum_{x,y\in X}e_{\tau_{y}(x),x}\otimes
e_{y,y},$ given in Remark 3.11. The explicit form of the 3-twist then is
${\cal G}_{012}=\sum_{\eta,x,y\in X}e_{\tau_{y}(\tau_{x}(\eta)),\eta}\otimes
e_{y,\sigma_{x}(y)}\otimes e_{\tau_{y}(x),\tau_{y}(x)}|_{C_{2}=0}$

The proofs of the statements above follow the same logic of the corresponding
proofs for ${\cal F}_{012}$.

The $(n+1)$-twist. The general $(n+1)$-twists may be derived by iteration
using the algebra co-product rules as identified by
$T_{0,12....n}=r_{0n}...r_{02}r_{01}$. For instance, recall the generalized
expressions ${\cal F}_{0,1...n}=\sum_{x_{1},...,x_{n},y_{n}\in
X}e_{\sigma_{x_{1}}(y_{1}),\sigma_{x_{1}}(y_{1})}\otimes
e_{x_{1},\tau_{y_{1}}(x_{1})}\otimes\ldots\otimes
e_{x_{n},\tau_{y_{n}}(x_{n})},$ subject to $~{}y_{m-1}=\sigma_{x_{m}}(y_{m}),$
$m\in\\{2,\ldots,n\\},$ and ${\cal F}_{01..n-1,n}=\sum_{x_{1},...,x_{n+1}\in
X}e_{x_{1},x_{1}}\otimes e_{x_{2},x_{2}}\otimes\ldots e_{x_{n},x_{n}}\otimes
e_{{\mathrm{X}},x_{n+1}},$ where
${\mathrm{X}}:=\sigma_{x_{1}}(\sigma_{x_{2}}(...(\sigma_{x_{n}}(x_{n+1}))...).$
We extract the $(n+1)$-twist via the generalized co-cycle condition (3.8) by
iteration: ${\cal F}_{012...n}=\sum_{x_{1},...,x_{n+1}\in
X}e_{x_{1},x_{1}}\otimes e_{\sigma_{x_{1}}(x_{2}),x_{2}}\otimes
e_{\sigma_{x_{1}}(\sigma_{x_{2}}(x_{3})),x_{3}}\otimes\ldots\otimes
e_{{\mathrm{X}},x_{n+1}}$, subject to the standard constraint
$\sigma_{x_{n}}(\sigma_{x_{n+1}}(y))=\sigma_{\sigma_{x_{n}}(x_{n+1})}(\sigma_{\tau_{x_{n+1}}(x_{n})}(y)),$
$\ x_{n},y\in X$.

######  Lemma 3.18.

Let ${\cal F}$ be an admissible twist, such that $r_{12}={\cal
F}^{-1}_{21}{\cal F}_{12},$ satisfies the YBE. Let also the Baxterized
solutions of the YBE $R(\lambda)=\lambda r+{\cal P},$
$\tilde{R}(\lambda)=\lambda{\mathbb{I}}+{\cal P}$ and
${\mathrm{T}}_{0}(\lambda)=R_{0N}(\lambda)\ldots R_{01}(\lambda),$
$~{}\tilde{\mathrm{T}}_{0}(\lambda)=\tilde{R}_{0N}(\lambda)\ldots\tilde{R}_{01}(\lambda),$
then

(3.40) $\tilde{\mathrm{T}}_{0}(\lambda)={\cal F}_{12..n0}\
{\mathrm{T}}_{0}(\lambda)\ {\cal F}^{-1}_{012..n}.$

###### Proof.

It suffices to prove that

(3.41) ${\cal F}_{12...n-1n0n+1...N}\ R_{0n}(\lambda)=\tilde{R}_{0n}(\lambda)\
{\cal F}_{12...n-10nn+1...N}$

then according to Propositions 3.5 and 3.7 expression (3.40) follows. Indeed,
for the admissible twist via Proposition 3.5, ${\cal F}_{12...n-1n0n+1...N}\
r_{0n}={\cal F}_{12...n-10nn+1...N},$ but also by the definition of the
permutation operator ${\cal P}$ we have
${\cal F}_{12...n-1n0n+1...N}\ {\cal P}_{0n}={\cal P}_{0n}\ {\cal
F}_{12...n-10nn+1...N}.$ Then according to the definition of $R,\ \tilde{R}$
we arrive at (3.41), which concludes our proof. ∎

###  3.3. Special case: Lyubashenko’s solution

We focus now on a special class of set-theoretic solutions of the YBE known as
Lyubashenko’s solutions [10] (see also [17] in relation to symmetric groups).
We first introduce this class of solutions and we show that they can be
expressed as simple twists recalling the results of [8]. We then move on to
show that these are admissible twists and we explicitly derive the associated
$n$-twists.

Let us recall Proposition 3.1 in [8]:

######  Proposition 3.19.

Let $\tau,\ \sigma:X\to X$ be isomorphisms, such that
$\sigma(\tau(x))=\tau(\sigma(x))=x$ and let ${\cal V}=\sum_{x\in
X}e_{x,\tau(x)}$ and ${\cal V}^{-1}=\sum_{x\in X}e_{\tau(x),x}$. Then any
solution of the braid YBE of the type

(3.42) $\check{r}=\sum_{x,y\in X}e_{x,\sigma(y)}\otimes e_{y,\tau(x)},$

can be obtained from the permutation operator ${\cal P}=\sum_{x,y\in
X}e_{x,y}\otimes e_{y,x}$ as

(3.43) $\check{r}=({\cal V}\otimes I){\cal P}({\cal V}^{-1}\otimes
I)=(I\otimes{\cal V}^{-1}){\cal P}(I\otimes{\cal V})$

We will now derive the $n$-twists associated to Lyubashenko’s solution and
show that these are admissible. We introduce in what follows two distinct
twists ${\cal F}$ and ${\cal G}$ compatible with the results of [8]. Let us
first introduce the twist ${\cal F}_{12}={\cal V}_{2},$ given the explicit
form ${\cal V}=\sum_{x\in X}e_{x,\tau(x)}$ we may rewrite ${\cal
F}=\sum_{x,y\in X}e_{x,x}\otimes e_{y,\tau(y)}$. It is instructive to compare
with the general set-theoretic twist, Proposition 3.10. Note that the terms at
each space now “decouple” due to the fact that $\sigma_{x}(y),\
\tau_{y}(x)\to\sigma(y),\ \tau(x)$.

Indeed, according to Proposition 3.19, ${\cal P}={\cal
V}_{2}\check{r}_{12}{\cal V}_{2}^{-1}$, where $\check{r}=\sum_{x,y\in
X}e_{x,\sigma(y)}\otimes e_{y,\tau(x)}$. Also, ${\cal F}$ is a Reshetikhin
type twist as it trivially satisfies the YBE, and consequently the co-product
structure is the one of the quasi-triangular Hopf algebra:

(3.44) ${\cal F}_{1,23}={\cal F}_{13}{\cal F}_{12},\quad{\cal F}_{12,3}={\cal
F}_{13}{\cal F}_{23}.$

By means of (3.44) and given that ${\cal F}_{12}={\cal V}_{2}$: we have:

(3.45) ${\cal F}_{1,23}={\cal V}_{2}{\cal V}_{3},\quad{\cal F}_{12,3}={\cal
V}_{3}^{2}.$

The co-cycle condition is satisfied and it is nothing but the Yang-Baxter
equation: ${\cal F}_{123}:={\cal F}_{12}{\cal F}_{12,3}$. We also derive by
iteration:

(3.46) ${\cal F}_{0,12...n}={\cal F}_{0n}\ldots{\cal F}_{01},\quad{\cal
F}_{12.n,0}={\cal F}_{10}{\cal F}_{20}\ldots{\cal F}_{n0},$

and the explicit expressions are given as

(3.47) ${\cal F}_{0,12..n}={\cal V}_{1}{\cal V}_{2}\ldots{\cal
V}_{n},\quad{\cal F}_{12..n,0}={\cal V}_{0}^{n}.$

The precise form of the $(n+1)$-twist is derived in the following Lemma.

######  Lemma 3.20.

Let ${\cal F}_{12}={\cal V}_{2}$, be the admissible twist for Lyubashenko’s
solution, then the $(n+1)$-twist is given as $\ {\cal
F}_{012..n}=\prod_{k=0}^{n}{\cal V}_{k}^{k}.$

###### Proof.

The proof is straightforward given the explicit forms of ${\cal F},$
expressions (3.47) and the generalized cocycle condition (3.8). ∎

Recall also that ${\cal V}=\sum_{x\in X}e_{x,\tau(x)},$ then we can express
the $(n+1)$-twist as

${\cal F}_{01...n}=\sum_{x_{1},...,x_{n+1}\in X}e_{x_{1},x_{1}}\otimes
e_{\sigma(x_{2}),x_{2}}\otimes e_{\sigma^{2}(x_{3}),x_{3}}\otimes\ldots\otimes
e_{\sigma^{n}(x_{n+1}),x_{n+1}}.$

We shall also need for the next Lemma the expression ${\cal F}_{12...n0}={\cal
V}_{0}^{n}{\cal V}_{n}^{n-1}\ldots{\cal V}_{3}^{2}{\cal V}_{2}.$

Given the explicit form of the $(n+1)$-twist from Lemma 3.20 we can show the
factorization of the monodromy matrix in a straightforward manner in the
following Lemma.

######  Lemma 3.21.

Let $T_{0,12..N}=r_{0N}...r_{01},$ where $r$ is Lyubashenko’s solution and
${\cal F}_{012..N}$, the $(N+1)$-twist as derived in Lemma 3.20, then $\ {\cal
F}^{-1}_{12..N0}\ {\cal F}_{012...N}=T_{0,12...N}.$

###### Proof.

The factorization can be checked by direct computation (recall in this case,
$r_{0n}={\cal V}_{0}^{-1}{\cal V}_{n}$), indeed

$\displaystyle{\cal F}_{12..N0}\ T_{0,12..N}\ {\cal F}^{-1}_{012...N}=$
$\displaystyle\Big{(}{\cal V}_{0}^{N}{\cal V}_{N}^{N-1}\ldots{\cal
V}_{3}^{2}{\cal V}_{2}\Big{)}\Big{(}{\cal V}_{0}^{-1}{\cal V}_{N}\ldots{\cal
V}_{0}^{-1}{\cal V}_{1}\Big{)}\Big{(}{\cal V}_{N}^{N}{\cal
V}_{N-1}^{N-1}\ldots{\cal V}_{2}^{2}{\cal
V}_{1}\Big{)}^{-1}=I^{\otimes(N+1)}.$

∎

######  Remark 3.22.

There is an alternative admissible twist for Lyubashenko’s solutions. Indeed,
${\cal G}_{12}={\cal V}_{1}^{-1},$ is an alternative admissible, Reshetikhin
type twist:

  1. (1)

${\cal G}_{12,3}={\cal G}_{13}{\cal G}_{23}$ and ${\cal G}_{1,23}={\cal
G}_{13}{\cal G}_{12}$.

  2. (2)

${\cal G}_{0,12...n}={\cal V}_{0}^{-n}$ and ${\cal G}_{12..n,0}={\cal
V}^{-1}_{1}{\cal V}^{-1}_{2}\ldots{\cal V}_{n}^{-1}.$

  3. (3)

Via the generalized co-cycle condition the alternative $(n+1)$-twist is
expressed as ${\cal G}_{01...n}=\prod_{k=0}^{n}{\cal V}_{k}^{-(n-k)}$.

Recalling also that ${\cal V}^{-1}=\sum_{x\in X}e_{x,\sigma(x)}$ we can write

${\cal G}_{01...n}=\sum_{x_{1},...,x_{n+1}\in
X}e_{\tau^{n}(x_{1}),x_{1}}\otimes
e_{\tau^{n-1}(x_{2}),x_{2}}\otimes\ldots\otimes e_{\tau(x_{n}),x_{n}}\otimes
e_{x_{n+1},x_{n+1}}.$

######  Remark 3.23.

The set-theoretic solution (2.2) is a representation of the Hecke algebra
$H_{N}(q=1)$ [7], hence the Baxterized solution can be expressed as in Remark
3.12. A $q$-analogue of Lyubashenko’s solution was derived in [8] via the
simple twist ${\cal V}=\sum_{x\in X}e_{\sigma(x),x}$. Indeed, if $g$ is the
${\mathfrak{U}}_{q}(\mathfrak{gl}_{\cal N})$ invariant representation of the
Hekce algebra $H_{N}(q)$ [22, 23] then so $g_{t}=({\cal V}\otimes I)g({\cal
V}^{-1}\otimes I)$ is provided that
$sgn(x-y)=sgn(\tau(x)-\tau(y))=sgn(\sigma(x)-\sigma(y))$ $\forall x,y\in X$
(note also that $[{\cal V}\otimes{\cal V},g_{t}]=0$). Then the Baxterized
solution $\check{R}_{t}=e^{\lambda}g_{t}–e^{-\lambda}g_{t}^{-1}$ gives a
twisted $\check{R}$-matrix associated to the
${\mathfrak{U}}_{q}(\mathfrak{gl}_{\cal N})$. The more general twist of
Proposition 3.10 can be also applied to the
${\mathfrak{U}}_{q}(\mathfrak{gl}_{\cal N})$ invariant solution $g$ of the
braid relation, subject to certain constraints, however this case will be
discussed in detail elsewhere.

### Curious observations.

This is a preliminary discussion motivating the next natural steps of our
investigation. Throughout this manuscript we have been focused on the
identification of set-theoretic twists and the issue of their admissibility in
the sense of Proposition 3.15. We have restricted our attention on the co-
product structure of the underlying quantum algebra and we have not discussed
the actions of the co-unit and antipode for set-theoretic solutions, which are
crucial in identifying the quantum algebra as a quasi-triangular Hopf algebra.

Let us briefly refer to the notions of co-unit and antipode for the quantum
algebra associated to Lyubashenko’s solution, which represents a simple, but
characteristic example of involutive, set-theoretic solution. Recall the
Lyubashenko solution can be expressed in the compact way $r:V\otimes V\to
V\otimes V$ ($V$ is the ${\cal N}$-dim vector space), such that $r={\cal
V}^{-1}\otimes{\cal V}$, where ${\cal V}=\sum_{x\in X}e_{x,\tau(x)}$ is a
group like element. From the definition $T_{0,12}=r_{02}r_{01}$ and recalling
the simple form of Lyubashenko’s solution we have $\Delta({\cal V})={\cal
V}\otimes{\cal V},$ indeed ${\cal V}$ is a group like element. We define the
co-unit: $\epsilon({\cal V})=\epsilon({\cal V})^{-1}=\epsilon(I)=1,$ then it
follows that $(\epsilon\otimes\mbox{id})\Delta({\cal
V})=(\mbox{id}\otimes\epsilon)\Delta({\cal V})={\cal V}.$ We also define the
antipode: $s({\cal V})={\cal V}^{-1},$ then
$m\big{(}(s\otimes\mbox{id})\Delta({\cal V})\big{)}=m\big{(}(\mbox{id}\otimes
s)\Delta({\cal V})\big{)}=I_{V}.$

We recall the Lyubashenko twists: ${\cal F}=I\otimes{\cal V}$ and ${\cal
G}={\cal V}^{-1}\otimes I$, we can then readily check, regardless the above
defined action of the co-unit, that $(\epsilon\otimes\mbox{id}){\cal F}={\cal
V},$ $\ (\mbox{id}\otimes\epsilon){\cal G}={\cal V}^{-1}$ and via
$\epsilon({\cal V})=\epsilon({\cal V}^{-1})=1,$ we conclude
$(\mbox{id}\otimes\epsilon){\cal F}=I,$ $\ (\epsilon\otimes\mbox{id}){\cal
G}=I.$ Even though the conditions $(\epsilon\otimes\mbox{id}){\cal F}=I,$
$(\mbox{id}\otimes\epsilon){\cal G}=I$ are now relaxed, the twists ${\cal F},\
{\cal G}$ are still admissible in the weaker sense of Proposition 3.15 and
Corollary 3.16. It is also interesting to present the action of the above
defined co-unit on Lyubashenkso’s $r$ matrix:
$(\epsilon\otimes\mbox{id})r={\cal V},$ $\ (\mbox{id}\otimes\epsilon)r={\cal
V}^{-1}$. This is an “uncommon” action of the co-unit, the origin of which is
the fact that both the set-theoretic $r$-matrix as well as the related twists
have no semi classical analogues, i.e. they can not be expressed as formal
series expansions with leading term being the identity.

Similar observations can be made for the Baxterized solutions coming from the
Lyubashenko $r$-matrix. Let $\tilde{R}(\lambda)=\lambda I_{V\otimes V}+{\cal
P}$ be the Yangian $R$-matrix where ${\cal P}=\sum_{x,y\in X}e_{x,y}\otimes
e_{y,x}$ is the permutation operator and $e_{x,y}$ are the generators of
$\mathfrak{gl}_{\cal N}$ in the ${\cal N}$-dimensional representation. We
focus for the sake of simplicity on the finite $\mathfrak{gl}_{\cal N}$
subalgebra of the Yangian; $\mathfrak{gl}_{\cal N}$ is a Hopf algebra with
counit $\epsilon(e_{x,y})=0$, $\epsilon(I)=1$, antipode $s(e_{x,y})=-e_{x,y}$
and co-product $\Delta(e_{x,y})=e_{x,y}\otimes I+I\otimes e_{x,y}.$ Moreover,
$\Delta(e_{x,y})\tilde{R}(\lambda)=\tilde{R}(\lambda)\Delta(e_{x,y}),$
$\forall x,y\in X$, recalling that $R(\lambda)={\cal
V}_{2}\tilde{R}_{12}(\lambda){\cal V}_{1}^{-1}={\cal
V}_{1}^{-1}\tilde{R}(\lambda){\cal V}_{2},$ we conclude that
$\Delta_{j}^{(op)}(e_{x,y})R(\lambda)=R(\lambda)\Delta_{j}(e_{x,y}),$
$j\in\\{1,\ 2\\},$ where the two types of twisted co-products are defined as
[8]: $\Delta_{1}(e_{x,y})=e_{x,y}\otimes I+I\otimes e_{\tau(x),\tau(y)}$ and
$\Delta_{2}(e_{x,y})=e_{\sigma(x),\sigma(y)}\otimes I+I\otimes e_{x,y},$
($\Delta_{2}(e_{x,y})={\cal V}\otimes{\cal
V}\cdot\Delta_{1}(e_{x,y})\cdot{\cal V}^{-1}\otimes{\cal V}^{-1}$).

Given the above twisted co-products it follows that
$(\epsilon\otimes\mbox{id})\Delta_{1}(e_{x,y})=e_{\tau(x),\tau(y)},$ $\
(\mbox{id}\otimes\epsilon)\Delta_{1}(e_{x,y})=e_{x,y}$ and
$(\epsilon\otimes\mbox{id})\Delta_{2}(e_{x,y})=e_{x,y},$ $\
(\mbox{id}\otimes\epsilon)\Delta_{1}(e_{x,y})=e_{\sigma(x),\sigma(y)}$ i.e.
the twisted co-products and co-unit do not satisfy the bialgebra axioms. In
addition, co-associativity for the deformed co-products is also an issue,
indeed it is easily shown using the explicit expressions for the co-products
$\Delta_{j}$:
$(\mbox{id}\otimes\Delta_{1})\Delta_{1}(e_{x,y})=\Phi^{-1}\cdot(\Delta_{1}\otimes\mbox{id})\Delta_{1}(e_{x,y})\cdot\Phi$
and
$(\mbox{id}\otimes\Delta_{2})\Delta_{2}(e_{x,y})=\hat{\Phi}^{-1}\cdot(\Delta_{2}\otimes\mbox{id})\Delta_{2}(e_{x,y})\cdot\hat{\Phi}$
where we define $\Phi=I\otimes I\otimes{\cal V}$ and $\hat{\Phi}={\cal
V}\otimes I\otimes I$, i.e. the co-associativity is now reduced to an almost
co-associativity. Although after applying the simple twists the underlying
algebra is still $\mathfrak{gl}_{\cal N}$ [8], strict co-associativity as well
as the axioms of the bialgebra involving the co-multiplication and co-unit are
not satisfied anymore.

We have briefly demonstrated the intriguing problem of characterizing the
quantum group associated to set-theoretic solutions as a bialgebra, using the
simple example of Lyubashenko’s solution. The problem for the general set-
theoretic solutions of the Yang-Baxter equation is studied in [9]. The general
case, also in relation to the Yangian, as well as the notion of the quantum
double will be addressed in detail elsewhere.

### Acknowledgments

I am grateful to A. Smoktunowicz for reading the manuscript and for useful
discussions and comments. Support from the EPSRC research grants EP/R009465/1
and EP/V008129/1 is also acknowledged.

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# Diamond-optic enhanced photon collection efficiency for sensing with
nitrogen-vacancy centers

Muhib Omar1,2  Andreas Conta2  Andreas Westerhoff2  Raphael Hasse2  Georgios
Chatzidrosos 1,2  Dmitry Budker1,2,3 and Arne Wickenbrock1,2  1Helmholtz-
Institut, GSI Helmholtzzentrum für Schwerionenforschung, Mainz, Germany
2Department of Physics, Mathematics and Computer Science, Johannes Gutenberg-
Universität Mainz, Mainz, Germany
3Department of Physics, University of California, Berkeley, Berkeley, CA,
United States<EMAIL_ADDRESS>
###### Abstract

We present a design to increase the amount of collected fluorescence emitted
by nitrogen-vacancy color centers in diamond used for quantum-sensing. An
improvement was measured in collected fluorescence when comparing oppositely
faced emitting surfaces by a factor of $3.8(1)$. This matches ray-tracing
simulation results. This design therefore improves on the shot noise limited
sensitivity in optical read-out based measurements of for instance magnetic
and electric fields, pressure, temperature and rotations.

††journal: oe††articletype: Research Article

##  1 Introduction

Nitrogen-vacancy centers (referred to as NV-centers in the following) are used
in various applications ranging from high-precision temperature measurements
[1], magnetic-field measurements in various modalitites, with [2, 3] or
without employing microwaves [4] or bias fields[5] for instance, electric-
field measurements[6] to quantum computing[7], gyroscopy [8] as well as bio-
sensing [9].

A NV-center is a point defect in diamond, where a pair of neighbouring carbon
atoms is replaced by a nitrogen atom and a vacancy. It is an atom-like system,
that can be in different charge states, called NV+, NV0 and NV-, where the
latter one is favoured for applications due to its optical spin read-out [10].
The NV- center has a total electron spin of 1, the corresponding electrons
being contributed by the NV-center nitrogen itself, the open bonds from the
carbon atoms and of another substitutional nitrogen atom in the lattice. The
electron spins can be optically pumped into one of the NV’s Zeeman sublevels
by illuminating the diamond with, for example, $532\text{\,}\mathrm{nm}$ laser
light driving transitions into the phonon broadened excited state. The spin
state can be read out by detecting the amount of (infra)red fluorescence
light, due to spin selective non-radiative transitions [10]. Driving microwave
transitions between the various spin states leads to observable changes in
fluorescence. Those can be used to measure magnetic fields over the respective
transition frequency shifts via the Zeeman effect .

A fundamental noise limit of fluorescence detection arises from photon shot
noise, depending on the amount of collected light which usually dominates over
spin-projection noise as another fundamental limit. Therefore, by increasing
the amount of collected light, the signal-to-noise ratio of such measurements
can be improved. Several different techniques have been developed to improve
photon-collection efficiency, in both single-NV setups [11, 13, 12] and
ensemble experiments [2, 14].

Figure 1: Diamond assembly images: (a) Sketch of the NV-bearing diamond
pyramid at the front (science diamond) dimensions on top of the diamond anvil.
(b) Photo of the science diamond, glued to the diamond anvil. c) Image of
fluorescence light collected from the back of the diamond anvil used for
alignment purposes and taken with a CCD camera. The circle in the center of
the cross is the apex of the fluorescence cone from the laser beam focal spot
and the four side beams arise due to the side reflections of the anvil. All
measures are in mm.

In this work, the diamond containing the NV centers, referred to as the
sensing diamond, was glued to a cone-shaped diamond piece, referred to as the
diamond anvil, which increases the amount of collected fluorescence light. The
sensing diamond was cut to direct side emitted fluorescence from the sensing
volume into the back direction via total internal reflection, see Fig. 1. The
curved back surface of the diamond anvil reduces losses due to total internal
reflection at the diamond to air interface. Detection with a photodiode
confirms an improvement factor of $3.8(1)$ expected from simulations compared
to the other exiting surface.

##  2 Sample preparation

The sensing diamond, a high-pressure high-temperature (HPHT) sample (Element
Six DNV-B14), is specified to contain 13 ppm nitrogen, 3.7 ppm NV--centers and
1.4 ppm NV0-centers. This specific sample is 13C-depleted (99.999% 12C). The
sample was irradiated with $5\text{\,}\mathrm{MeV}$ electrons at a dose of
$2\text{\times}{10}^{19}\text{\,}\mathrm{c}\mathrm{m}^{-2}$ and then annealed
at $700\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for eight hours. Its measured
minimal linewidth in a pulsed optically detected magnetic resonance (ODMR)
experiment is around $300\text{\,}\mathrm{kHz}$.

The shape of the diamond anvil and sensing diamond pieces was optimised using
the COMSOL Multiphysics software. The simulations were used to evaluate the
improvement in fluorescence collection between the back and front side.

The science diamond is a trapezoid with a back surface being a square
$0.5\text{\,}\mathrm{mm}$ on the side, a height of $0.18\text{\,}\mathrm{mm}$,
and the upper square surface being $0.15\text{\,}\mathrm{mm}$ on the side, see
Fig. 1. The base angle for this shape is close to 45 degrees to match with the
single-crystal diamond anvil manufactured by Dutch Diamond Technologies. This
limits the angular distribution of about 90% of rays exiting the diamond
construction to below 45 degrees with respect to the symmetry axis, see Fig.
2. This means 90% of the light can be picked up by a lens with numerical
aperture of 0.7. A very weak requirement. The two diamond pieces were joined
with a thin layer of Norland Optical Adhesive 170 with refractive index of 1.7
(the highest-index material that we could find) applied between the anvil and
the back surface of the sensing diamond while pressing the pieces together.
Effects as for instance etaloning due to a significant glue layer thickness
were not observed.

In the COMSOL simulation within the ray-tracing module, the number of
collected rays were given a cylindrical distribution of ray sources inside the
sensing diamond to mimic the excitation volume shape by the laser light inside
the diamond. Three $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spaced point
sources arranged along the symmetry axes of the sensing diamond emitted
isotropically each 2000 rays. The ratio of collected rays between the front
and back side on a photodiode using an 8 mm focal length, 12.7 mm diameter
aspheric condenser lens was 3.8.

The diamond was mounted on a custom made peek mount during the experiment.
Trying to test thermal durability of the glue joint, we applied around 1.8 W
of green laser light in a 0.9 mm diameter beam focused using 8 mm focal length
lens for around 10 s on the sensing diamond from the back side. No degradation
of the diamond optical assembly was observed at temperatures at which the peek
material started to deform which are estimated to be around 150 °C based on
its glass transition temperature [15].

Figure 2: Simulated fractional cumulative angular ray distribution as a
function of $\alpha$, the ray angle with respect to the symmetry axis of the
diamond anvil with the sensing diamond. The cumulative ray fraction per single
side collection is indicated as fraction of total number of emitted rays per
side. The dashed red line indicates the numerical aperture of the collecting
lens used in this note, the grey one the anvil opening angle of 45 degrees.

##  3 Characterisation measurements

To verify the simulation results we built a setup to measure the amount of
fluorescence light collected from the front and back sides of the diamond
simultaneously.

Figure 3: Experimental setup: the $532\text{\,}\mathrm{nm}$ laser light is
focused on a NV-center doped sensing diamond, attached to a diamond anvil. The
emitted light is focused onto the same model of photodiodes on each side. A
dichroic mirror is used to block the reflected green light going towards the
first photodiode, which is used to to record the intensity of the fluorescence
light. An interference filter is employed to reject the laser light and
transmit the fluorescence light. The right photodiode and lens were replaced
with a CCD camera and a longer focal length lens respectively to capture
images used to align the light beam with respect to the diamond.

###  3.1 Experimental setup

The setup is sketched in Fig. 3. A $532\text{\,}\mathrm{nm}$ laser beam was
focused into the sensing diamond using a plano-convex lens with a focal length
of $f=$ $8\text{\,}\mathrm{mm}$. Behind the diamond we placed initially
another lens and a notch filter to separate out green light from the
(infra)red fluorescence light emitted from the diamond detected with a charge-
coupled device (CCD) camera. That way we were able to verify that the diamond
was well centred illuminated with the $532\text{\,}\mathrm{nm}$ light. The
camera was positioned on the back side producing the characteristic cross
shape shown in Fig. 1 (c). This shape originates from reflections of the side
surfaces of the sensing diamond and allows for precise positioning of the
diamond relative to the laser beam using a XYZ-stage.

After alignment, the optics at the back side were replaced with the same type
of aspheric condenser lens with $f$=$8\text{\,}\mathrm{mm}$ focal length, a
notch filter and a photodiode. The fluorescence was compared in both front and
back direction simultaneously. Integrated over the expected fluorescence
spectrum the notch filter (Thorlabs NF 533-17) transmits about 2% more than
the dichroic mirror (Thorlabs DMLP-567 ), negligible within the measurement
error.

Figure 4: Comparison of the signal improvement due to the higher photon
collection efficiency exemplified by an optically detected magnetic resonance
spectra, both normalised to the back surface peak signal value.

###  3.2 Measurements

Measuring on both sides simultaneously, for five laser powers equally spaced
between 50 and 150 mW gave a mean increase of collected fluorescence light by
a factor of $3.8(1)$ between the back and front side. Next, a magnetic field
was applied with Helmholtz coils and we used microwaves to obtain ODMR spectra
of the NV centers to visualize the difference, see Fig. 4.

##  4 Conclusion

We described a design to improve the amount of collected fluorescence light
emitted by a nitrogen-vacancy center ensemble in diamond. We were able to
experimentally measure an increase by a factor of $3.8(1)$ between the
improved design (back) with respect to the not improved opposing facet
(front). This increase is supported by ray tracing simulations. An additional
feature of the design is the improved angular distribution of the
fluorescence. It would allow for over 90% of the emitted fluorescence to be
collected by a lens with a numerical aperture bigger than 0.7. These lenses
are widely available. In sensing applications relying on the collected
fluorescence this improvement results in a lowered shot noise limit by a
factor of nearly 2. Further improvement in overall light collection are
possible for example deploying reflective coating on the front surface and
anti-reflective coating on the back surface. Including the coating and
neglecting losses this optic then allows to collect all emitted photons, which
amounts to an additional increase of more than 40%.

##  5 Funding

This work was supported by the European Commission’s Horizon Europe Framework
Program under the Research and Innovation Action MUQUABIS GA no.10107054 and
by the German Federal Ministry of Education and Research (BMBF) within the
MILIQUANT project no. 13N15062 and DIAQNOS project no. 13N16455.

##  6 Acknowledgements

We thank Dr. Till Lenz, Joseph Shaji Rebeirro and Omkar Dhungel for the many
and fruitful discussions concerning this project.

##  7 Disclosures

The authors declare no conflicts of interest.

##  8 Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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