# Lusztig’s Root Vectors and a Dolbeault Complex for the $A$-Series Full
Quantum Flag Manifolds

Réamonn Ó Buachalla  Mathematical Institute of Charles University, Sokolovská
83, Prague, Czech Republic<EMAIL_ADDRESS>and  Petr Somberg
Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech
Republic<EMAIL_ADDRESS>I ndíl chuimhne ar Sheán Ó Buachalla

###### Abstract.

For the Drinfeld–Jimbo quantum enveloping algebra
$U_{q}(\mathfrak{sl}_{n+1})$, we show that the span of Lusztig’s positive root
vectors, with respect to Littlemann’s nice reduced decompositions of the
longest element of the Weyl group, form quantum tangent spaces for the full
quantum flag manifold $\mathcal{O}_{q}(\mathrm{F}_{n+1})$. The associated
differential calculi are direct $q$-deformations of the anti-holomorphic
Dolbeault complex of the classical full flag manifold $\mathrm{F}_{n+1}$. As
an application we establish a quantum Borel–Weil theorem for the
$A_{n}$-series full quantum flag manifold, giving a noncommutative
differential geometric realisation of all the finite-dimensional type-$1$
irreducible representations of $U_{q}(\mathfrak{sl}_{n+1})$. Restricting this
differential calculus to the quantum Grassmannians is shown to reproduce the
celebrated Heckenberger–Kolb anti-holomorphic Dolbeault complex. Lusztig’s
positive root vectors for non-nice decompositions of the longest element of
the Weyl group are examined for low orders, and are exhibited to either not
give tangents spaces, or to produce differential calculi of non-classical
dimension.

RÓB is supported by the Charles University PRIMUS grant _Spectral
Noncommutative Geometry of Quantum Flag Manifolds_ PRIMUS/21/SCI/026. PS is
supported by the GAČR grant _GACR 22-00091S_. This article is based upon work
from COST Action 21109 CaLISTA, supported by COST (European Cooperation in
Science and Technology). www.cost.eu

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of the Paper
  2. 2 Preliminaries
    1. 2.1 Covariant Differential Calculi over Hopf algebras
    2. 2.2 Tangent Spaces and Two-Sided Hopf Modules
    3. 2.3 Generating Relations of the Maximal Prolongation
  3. 3 A First-Order Differential Calculus for $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$
    1. 3.1 The Lusztig Root Vectors for $U_{q}(\mathfrak{sl}_{n+1})$
    2. 3.2 The Module Structure of the Cotangent Space $\Lambda^{(0,1)}_{q}$
    3. 3.3 Relations of the Maximal Prolongation
    4. 3.4 Some Low Rank Examples
    5. 3.5 A Filtration for the Quantum Exterior Algebra
    6. 3.6 The Frobenius and Koszul Properties
  4. 4 The Heckenberger–Kolb Dolbeault Complex for the Quantum Grassmannians
    1. 4.1 Some Preliminaries on Quantum Homogeneous Spaces
    2. 4.2 Some Remarks on Quantum Homogeneous Tangent Spaces
    3. 4.3 Lusztig Differential Calculi for the $A$-Series Quantum Flag Manifolds
    4. 4.4 The Heckenberger–Kolb FODC for the Quantum Grassmannians
    5. 4.5 Some General Lemmas
    6. 4.6 Embedding the Higher-Order Forms
  5. 5 The Lusztig DC for the Full Quantum Flag Manifolds
    1. 5.1 Some Details on Full Quantum Flag Manifolds
    2. 5.2 A Higher Order Extension of Majid’s Framing Theorem
    3. 5.3 The Lusztig Differential Calculus of $\mathcal{O}_{q}(\mathrm{F}_{n+1})$
    4. 5.4 The Full Lusztig DC and the Sub-Category ${}^{A}_{B}\mathrm{Mod}_{0}$
    5. 5.5 Some Examples and Observations
  6. 6 A Borel–Weil Theorem for the Full Quantum Flag Manifolds
    1. 6.1 Connections and Quantum Principal Bundles
    2. 6.2 Holomorphic Structures and Principal Connections
    3. 6.3 A Borel–Weil Theorem for $\mathcal{O}_{q}(\mathrm{F}_{n+1})$
    4. 6.4 Quantum Homogeneous Coordinate Rings
  7. 7 Outside the Nice Setting
    1. 7.1 The Opposite Involution, Reduced Decompositions, and DC
    2. 7.2 The Case of $U_{q}(\mathfrak{sl}_{4})$
    3. 7.3 The Case of $U_{q}(\mathfrak{sl}_{5})$
    4. 7.4 Some Conjectures
    5. 7.5 A General Class of Tangent Spaces for $U_{q}(\mathfrak{sl}_{3})$
  8. A The $A$-Series root vectors
    1. A.1 The $A_{n}$ Root System
    2. A.2 The Drinfeld–Jimbo Quantised Enveloping Algebra $U_{q}(\mathfrak{sl}_{n+1})$
    3. A.3 The Quantum Coordinate Algebra $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$
    4. A.4 Lusztig’s Root Vectors
    5. A.5 The Explicit Calculation
    6. A.6 The Opposite Involution of $U_{q}(\mathfrak{sl}_{n+1})$
  9. B Some Remarks on Filtered Algebras

##  1\. Introduction

The Heckenberger–Kolb differential calculi for the irreducible quantum flag
manifolds are among the most important and well-studied structures in the
noncommutative geometry of quantum groups. They are an essentially unique
covariant $q$-deformation of the de Rham complex of the classical flag
manifolds of Hermitian symmetric type, a striking phenomemon given the many
difficulties in formulating a comprehensive theory of noncommutative geometry
for Drinfeld–Jimbo quantum groups. Moreover, as has become increasingly clear
in recent years, these calculi come with an extremely rich $q$-deformed Kähler
geometry. They admit direct noncommutative generalisations of Lefschetz
decomposition and the Kähler identities [31, 29], of the Borel–Weil theorem
[5, 4], and Kodaira vanishing [32]. This has allowed for the construction of
Dolbeault–Dirac Fredholm operators from noncommutative complex geometric
arguments [8], and for the construction of spectral triples, in the sense of
Connes, for special cases such as quantum projective space [9] and the first
odd quantum quadric [11].

It is natural to ask is what happens outside the irreducible setting, which is
to say, when the classical flag manifold is no longer of Hermitian symmetric
type. In this more general setting, quantum flag manifolds exhibit a richer
and more challenging structure. Even in the classical setting, the holomorphic
and anti-holomorphic tangent spaces of a non-irreducible flag manifold are no
longer irreducible as modules over the corresponding Levi subalgebra. For
example, the space of one forms of the full flag manifold decomposes into a
direct sum of line bundles. Moreover, in the quantum setting, the dual
coalgebras of the non-irreducible quantum flag manifolds contain far fewer
twisted primitive elements. Even though the Heckenberger–Kolb calculi appeared
over twenty years ago, up until now, the question of noncommutative
differential structures for non-irreducible examples has received very little
attention. There are, to our knowledge, just two studies: First is the quantum
BGG sequence of Heckenberger and Kolb [20]. Second is the work of Yuncken and
Voigt [45] on the noncommutative geometry of the full quantum flag manifold of
$\mathcal{O}_{q}(\mathrm{SU}_{3})$, which resulted in a proof of the quantum
Baum–Connes conjecture for the discrete quantum dual of
$\mathcal{O}_{q}(\mathrm{SU}_{3})$.

Here we adopt an approach to this problem based on Lusztig’s celebrated root
vectors. Classically the tangent space of the full flag manifold is isomorphic
to the nilpotent Lie algebra $\mathfrak{n}\subseteq\mathfrak{g}$. Formulating
a quantum analogue of the inclusion $\mathfrak{g}\hookrightarrow
U(\mathfrak{g})$ is a subtle question, see for example the approach of Majid
[27]. In [26], Lusztig showed that the embedding of the nilpotent subalgebra
$\mathfrak{n}\hookrightarrow U(\mathfrak{g})$ has a direct quantum analogue
for each choice of reduced decomposition of the longest element of the Weyl
group of $\mathfrak{g}$. These quantum root vectors were originally introduced
to prove a quantum generalisation of the classical Poincaré–Birkhoff–Witt
theorem, and would later find many striking applications, most notably the
theory of canonical bases [26]. Our approach to extending the anti-holomorphic
Heckenberger–Kolb construction starts by taking the span of Lusztig’s positive
root vectors as a tangent space for the full quantum flag, and then
constructing a differential calculus from it. This allows us to make novel
connections between the structural theory of Drinfeld–Jimbo quantum groups and
their noncommutative differential geometry.

In this paper we focus on the $A$-series Drinfeld–Jimbo quantum groups, and
the reduced decomposition $\mathbf{j}$ given by

(1) $\displaystyle w_{0}=(s_{n}s_{n-1}\cdots s_{1})(s_{n}s_{n-1}\cdots
s_{2})\cdots(s_{n}s_{n-1})s_{n}.$

This decomposition is one of the two nice decompositions in the sense of
Littlemann [24, §5]. We show that the associated space of positive Lusztig
root vectors forms a quantum tangent space for the full quantum flag manifold.
The associated covariant differential calculus is a direct $q$-deformation of
the anti-holomorphic subcomplex of the classical $A$-series full quantum flag
manifold, in particular it has classical dimension. Its quantum exterior
algebra admits a concise generator and relation presentation in terms of the
$A$-series root system. The quantum exterior algebra is moreover a Frobenius
algebra. Notably, its Nakayama automorphism is _not_ $q$-deformed. Restricting
this differential calculus to the partial quantum flag manifolds then gives a
differential calculus for each of them. In particular, for the quantum
Grassmannians, we recover the anti-holomorphic subcomplex of the
Heckenberger–Kolb differential calculus.

In order to explore and test our new construction, we prove a direct
$q$-deformation of the Borel–Weil theorem for the full flag manifold. This
gives noncommutative geometric realisations of all the type-$1$ finite-
dimensional representations of $U_{q}(\mathfrak{sl}_{n+1})$, extending
previous work for the irreducible quantum flag case [5, 4]. The proof requires
the introduction of a quantum principal bundle presentation, in the sense of
Majid and Brzezinski, of our differential calculus. Moreover, it presents the
relative line modules over $\mathcal{O}_{q}(\mathrm{F}_{n+1})$ as
noncommutative holomorphic line modules in the sense of Beggs and Majid [1,
§7].

For low-dimensional cases, we also examine the behaviour of the positive
Lusztig roots vectors associated to the other reduced reduced decompositions.
We see that for $A_{3}$, all $8$ commutator classes of reduced decompositions
give tangent spaces. However, apart from the decompositon dual to (1) (that is
the decomposition given by the opposite automorphism of the $A$-series Dynkin
diagram) the associated differential calculi have non-classical dimension.
Moreover, for the case of $A_{4}$, there are reduced decompositions that are
not even tangent spaces. This motivates the conjecture that for the
$A$-series, Littlemann’s nice reduced decompositions are the only
decompositions with a well-behaved noncommutative geometry.

This work leads naturally to a number of future topics. First is the question
of a $q$-deformation of the entire Dolbeault double complex, along with its
classical Kähler geometry. This will be addressed in subsequent work using
Lusztig’s negative root vectors. Subsequently, the $B,C$, and $D$ series will
be treated following the same program. Ultimately, it is hoped that this will
show the way to an understanding of the noncommutative geometry of
Drinfeld–Jimbo coordinate algebras themselves.

###  1.1. Summary of the Paper

The paper is organised as follows: In §2 we recall some basic material about
covariant differential calculi over Hopf algebras and their associated tangent
spaces, as well as the maximal prolongation of covariant first-order
differential calculi.

In §3 we treat the space of positive root vectors associated to the reduced
decomposition of the longest element of the Weyl group given in (1), and show
that their span forms a quantum tangent space for
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. We then determine the module structure
of the dual cotangent space, and give a full set of relations for the maximal
prolongation of the associated differential calculus
$\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})$. We also present a number of
instructive low-dimensional examples. We then examine the algebraic properties
of the maximal prolongation, and show that it has classical dimension by
examining the associated graded algebra of a canonical filtration on
$\Lambda^{(0,\bullet)}_{q}$. We also show that the graded algebra
$\Lambda^{(0,\bullet)}_{q}$ is a Frobenius algebra and show that the
associated Nakayama automorphism is of classical type.

In §4 we begin by recalling some basic results on quantum homogeneous spaces
and their tangent spaces. We then examine the restriction of
$\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})$ to the quantum Grassmannians,
showing that we recover the anti-holomorphic Heckenberger–Kolb differential
calculus.

In §5 we examine the restriction of
$\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})$ to the full quantum flag
manifold $\mathcal{O}_{q}(\mathrm{F}_{n+1})$. We show that this differential
calculus has classical dimension, and present it as a direct sum of line
bundles over $\mathcal{O}_{q}(\mathrm{F}_{n+1})$. Notably, we show that its
right module structure is not the trivial one, as is the case for the
Heckenberger–Kolb calculus, meaning that it lies outside the subcategory of
relative Hopf modules to which the simple monoidal version of Takeuchi’s
equivalence applies.

In §6 we prove a direct $q$-deformation of the Borel–Weil theorem for the
classial full flag manifold, giving noncommutative geometric realisations of
all the type $I$ finite-dimensional irreducible representations of
$U_{q}(\mathfrak{sl}_{n+1})$.

In §7, for the special cases of $U_{q}(\mathfrak{sl}_{4})$ and
$U_{q}(\mathfrak{sl}_{5})$, we examine the situation for reduced
decompositions other than Littlemann’s nice decompositions. In all cases
examined, we see that the space of root vectors either do not give a tangent
space, or the associated differential calculus does not have classical
dimension. Motivated by this, we make some conjectures about the general
$U_{q}(\mathfrak{sl}_{n+1})$ situation.

We finish with two appendices. In the first we present necessary details on
Drinfeld–Jimbo quantum groups and Lusztig’s root vectors. In the second we
make some comments about filtered algebras and their associated graded
algebras.

#### Acknowledgements:

We would like to thank Arnab Bhattacharjee, Alessandro Carotenuto, Andrey
Krutov, Karen Strung, and Bart Vlaar, for many useful discussions. We would
also like to thanks Nicolas M. Thiery and Nathan Williams for providing
references related to commutator classes of reduced decompositions.

##  2\. Preliminaries

In this section we recall some basic material about covariant differential
calculi over Hopf algebras and their associated tangent spaces. We use
Sweedler notation, denote by $\Delta,\varepsilon$, and $S$ the coproduct,
counit, and antipode of a Hopf algebra respectively. Throughout the paper, all
algebras are over $\mathbb{C}$ and assumed to be unital, all unadorned tensor
products are over $\mathbb{C}$, and all Hopf algebras are assumed to have
bijective antipodes.

###  2.1. Covariant Differential Calculi over Hopf algebras

A differential calculus, or a _dc_ , is a differential graded algebra (dga)

$\big{(}\Omega^{\bullet}\simeq\bigoplus_{k\in\mathbb{Z}_{\geq
0}}\Omega^{k},\mathrm{d}\big{)}$

which is generated as an algebra by the elements $a,\mathrm{d}b$, for
$a,b\in\Omega^{0}$. When no confusion arises, we denote the dc by
$\Omega^{\bullet}$, omitting the exterior derivative $\mathrm{d}$. We denote
the degree of a homogeneous element $\omega\in\Omega^{\bullet}$ by $|\omega|$.
For a given algebra $B$, a differential calculus over $B$ is a differential
calculus such that $B=\Omega^{0}$. A differential calculus is said to be of
total degree $m\in\mathbb{Z}_{\geq 0}$ if $\Omega^{m}\neq 0$, and
$\Omega^{k}=0$, for all $k>m$.

A first-order differential calculus (fodc) over an algebra $B$ is a pair
$(\Omega^{1}(B),\mathrm{d})$, where $\Omega^{1}(B)$ is a $B$-bimodule and
$\mathrm{d}:B\to\Omega^{1}(B)$ is a derivation such that $\Omega^{1}$ is
generated as a left (or equivalently right) $B$-module by those elements of
the form $\mathrm{d}b$, for $b\in B$. We say that a differential calculus
$(\Gamma^{\bullet},\mathrm{d}_{\Gamma})$ extends a first-order calculus
$(\Omega^{1},\mathrm{d}_{\Omega})$ if there exists a bimodule isomorphism
$\varphi:\Omega^{1}\to\Gamma^{1}$ such that
$\mathrm{d}_{\Gamma}=\varphi\circ\mathrm{d}_{\Omega}$. It can be shown [30,
§2.5] that any first-order calculus admits an extension $\Omega^{\bullet}$
which is _maximal_ , in the sense that there exists a unique differential map
from $\Omega^{\bullet}$ onto any other extension of $\Omega^{1}$. We call this
extension the maximal prolongation of the first-order calculus.

For $A$ a Hopf algebra, a dc $\Omega^{\bullet}$ over a left $A$-comodule
algebra $(P,\Delta_{L})$ is said to be _left covariant_ if the coaction
$\Delta_{L}:P\to A\otimes P$ extends to a (necessarily unique) coaction
$\Delta_{L}:\Omega^{\bullet}\to A\otimes\Omega^{\bullet}$, giving
$\Omega^{\bullet}$ the structure of a left $A$-comodule algebra, such that
$\mathrm{d}$ is a left $A$-comodule map. A right covariant dc over a right
$A$-comodule algebra is defined similarly.

###  2.2. Tangent Spaces and Two-Sided Hopf Modules

A _tangent space_ for a Hopf algebra $A$ is a linear subspace $T$ of the dual
Hopf algebra $A^{\circ}$ such that $X(1)=0$, for all $X\in T$, and
$T\oplus\mathbb{C}1$ is a left coideal of $A^{\circ}$. For any tangent space
$T$, a right $A$-ideal of $A^{+}:=\mathrm{ker}(\varepsilon)$ is given by

(2) $\displaystyle I:=\Big{\\{}x\in A^{+}\,|\,X(x)=0,\textrm{ for all }X\in
T\Big{\\}},$

meaning that the quotient $\Lambda^{1}(A):=A^{+}/I$ is naturally an object in
the category $\mathrm{Mod}_{A}$ of right $A$-modules. We call $\Lambda^{1}(A)$
the _cotangent space_ of the fodc.

Consider next the category ${}^{A}_{A}\mathrm{Mod}_{A}$: This has as objects
left $A$-comodules ${\mathcal{F}}$, endowed with an $A$-bimodule structure,
such that the comodule structure map is an $A$-bimodule map with respect to
the diagonal $A$-bimodule structure on $A\otimes{\mathcal{F}}$. Its morphisms
are left $A$-comodule, $A$-bimodule, maps. We have a functor

$\displaystyle A\otimes-:\mathrm{Mod}_{A}\to{}^{A}_{A}\mathrm{Mod}_{A},$
$\displaystyle V\mapsto A\otimes V,$

where the left $A$-module, left $A$-comodule, structure of $A\otimes V$ is
given by the first tensor factor, and the right $A$-module structure is given
by the diagonal action. In the other direction, we have a functor

$\displaystyle F:{}^{A}_{A}\mathrm{Mod}_{A}\to\mathrm{Mod}_{A},$
$\displaystyle{\mathcal{F}}\mapsto{\mathcal{F}}/A^{+}{\mathcal{F}}.$

This determines an adjoint equivalence between the two categories, with unit
and counit

$\displaystyle\mathrm{U}:{\mathcal{M}}\to A\otimes F({\mathcal{M}}),$
$\displaystyle g\mapsto g_{(1)}\otimes[g_{(0)}],$
$\displaystyle\mathrm{C}:F(A\otimes V)\to V,$ $\displaystyle a\otimes
v\mapsto\varepsilon(a)v.$

This equivalence is known as the _fundamental theorem of two-sided Hopf
modules_ , see [38, Theorem 5.7], or [42, §4.1], for more details. The
category ${}^{A}_{A}\mathrm{Mod}_{A}$ has a monoidal structure given by the
usual tensor products of $A$-bimodules and left $A$-comodules. Moreover,
$\mathrm{Mod}_{A}$ has a monoidal structure given by the tensor product of
vector spaces endowed with the diagonal $A$-module structure. With respect to
these two monoidal structures, the natural transformation with components

(3) $\displaystyle\mu:F({\mathcal{M}})\otimes F({\mathcal{N}})\to
F({\mathcal{M}}\otimes_{A}{\mathcal{N}}),$
$\displaystyle[m]\otimes[n]\mapsto[mS(n_{(-1)})\otimes n_{(0)}].$

gives $F$ the structure of a monoidal functor. Note that the inverse of $\mu$
is given by

$\displaystyle\mu^{-1}:F({\mathcal{M}}\otimes{\mathcal{N}})\to
F({\mathcal{M}})\otimes_{A}F({\mathcal{N}}),$ $\displaystyle[m\otimes
n]\mapsto[mn_{(-1)}]\otimes[n_{(0)}]$

Consider now the object

$\displaystyle{}^{A}_{A}\mathrm{Mod}_{A}\ni\Omega^{1}(A):=A\otimes\Lambda^{1}(A).$

If $\\{X_{i}\\}_{i=1}^{n}$ is a basis for $T$, and $\\{e_{i}\\}_{i=1}^{n}$ is
the dual basis of $\Lambda^{1}(A)$, then the map

(4) $\displaystyle\mathrm{d}:A\to\Omega^{1}(A),$ $\displaystyle
a\mapsto[a_{(1)}]\otimes[a_{(2)}^{+}]=\sum_{i=1}^{n}(X_{i}\triangleright
a)\otimes e_{i}$

is a derivation, and the pair $(\Omega^{1}(A),\mathrm{d})$ is a left
$A$-covariant fodc over $A$. Moreover, it follows from the theorem of two-
sided Hopf modules that all fodc, which are finitely generated as left
$A$-modules, arise from a finite-dimensional tangent space. Indeed, this gives
a bijective correspondence between isomorphism classes of finite-dimensional
tangent spaces and finitely-generated left-covariant fodc [46].

Let $W$ be a Hopf subalgebra of $A^{\circ}$, and denote by
$\pi:(A^{\circ})^{\circ}\to W^{\circ}$ the Hopf algebra map given by
restriction of domains. Considering $A$ as contained in $(A^{\circ})^{\circ}$,
we denote $K:=\pi(A)$, and call $K$ the _quantum subgroup_ associated to $W$.
Now if a tangent space $T\subseteq A^{\circ}$ is invariant under the adjoint
action of $W$, that is, if $WT\subseteq T$, then the corresponding ideal
$I\subseteq A^{+}$ is a subcomodule with respect to the _relative adjoint
coaction_

$\displaystyle\mathrm{Ad_{R,\pi}}:A^{+}\to A^{+}\otimes K,$ $\displaystyle
a\mapsto a_{(2)}\otimes\pi(S(a_{(1)})a_{(3)}).$

This endows $\Lambda^{1}(A)$ with a right $K$-coaction, which in turn gives
the associated dc $\Omega^{\bullet}$ the structure of a right $K$-covariant
dc, defined by the tensor product $K$-coaction. Moreover, via the dual pairing
$W\times K\to\mathbb{C}$, both $\Lambda^{1}(A)$ and $\Omega^{\bullet}$ have
left $W$-actions dual to their right $K$-coactions.

Just as for any covariant fodc, this right $K$-coaction extends to a right
$K$-coaction on $\Omega^{\bullet}$, the maximal prolongation of $\Omega^{1}$.
This gives $\Omega^{\bullet}$ the structure of a right $K$-comodule algebra.
Together the left $A$-coaction and the right $K$-coaction give
$\Omega^{\bullet}$ the structure of an $(A,K)$-bicomodule. This means that the
space of left $A$-coinvariant forms $1\otimes\Lambda^{\bullet}$ is a right
$K$-subcomodule of $\Omega^{\bullet}$. In particular, it is a right
$K$-comodule algebra.

###  2.3. Generating Relations of the Maximal Prolongation

An explicit presentation of the maximal prolongation of a left $A$-covariant
fodc $\Omega^{1}(A)$ over a Hopf algebra $A$ is given as follows: For
$I\subseteq A^{+}$ the classifying ideal of the fodc, and $\Lambda^{1}(A)$ the
cotangent space, consider the subspace

$\displaystyle
I^{(2)}:=\left\\{\omega(x):=[x_{(1)}^{+}]\otimes[x_{(2)}^{+}]\leavevmode\nobreak\
|\leavevmode\nobreak\ x\in
I\right\\}\subseteq\Lambda^{1}(A)\otimes\Lambda^{1}(A).$

From the tensor algebra $\mathcal{T}(\Lambda^{1}(A))$ of $\Lambda^{1}(A)$, we
construct the $\mathbb{Z}_{\geq 0}$-graded algebra

$\displaystyle\Lambda^{\bullet}(A):=\bigoplus_{k\in\mathbb{Z}_{\geq
0}}\Lambda^{k}(A):=\mathcal{T}\big{(}\Lambda^{1}(A)\big{)}/\langle
I^{(2)}\rangle,$

which we call the _quantum exterior algebra_ of $\Omega^{1}(A)$, and whose
multiplication we denote by $\wedge$. It follows from the fundamental theorem
of Hopf modules, and (4), that an isomorphism between $F(\Omega^{k}(A))$ and
$\Lambda^{k}(A)$, for $k\in\mathbb{Z}_{\geq 0}$, is determined by

$\displaystyle[\mathrm{d}a_{1}\wedge\cdots\wedge\mathrm{d}a_{k}]\leavevmode\nobreak\
\mapsto\leavevmode\nobreak\
\bigwedge_{i-1}^{k}\,[(a_{i})^{+}_{(i)}(a_{(i+1)})_{(i)}\cdots(a_{(k)})_{(i)}],$

where we have denoted $(a_{1})_{(1)}:=a_{1}$. See [21] for further details, or
[30] for the more general quantum homogeneous space case.

##  3\. A First-Order Differential Calculus for
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$

In this section we show that, for a distinguished decomposition of the longest
element of the Weyl group, Lusztig’s root vectors give a tangent space for
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. The maximal prolongation of the
associated fodc is then described, and shown to have classical dimension. We
also show that quantum exterior algebra $\Lambda^{(0,\bullet)}_{q}$ has the
structure of a Frobenius algebra and prove that its Nakayama automorphism is
of classical type. Throughout, low rank examples are treated in detail.
Throughout we use the notation set in Appendix A. Moreover, we find it
convenient to identify $\Delta^{+}$ with the set of natural number pairs
$(i,j)$ for which $\alpha_{ij}$ is a positive root.

###  3.1. The Lusztig Root Vectors for $U_{q}(\mathfrak{sl}_{n+1})$

Consider the decomposition $\mathbf{j}$ of the longest element of the Weyl
group $w_{0}\in S_{n+1}$:

$\displaystyle w_{0}=(s_{n}s_{n-1}\cdots s_{1})(s_{n}s_{n-1}\cdots
s_{2})\cdots(s_{n}s_{n-1})s_{n},$

where $s_{1},\dots,s_{n}$ denote the standard generators of $S_{n+1}$. As
calculated in Appendix A, the root vectors corresponding to this decomposition
are given explicitly by

$\displaystyle
E_{ji}:=[E_{j-1},[E_{j-2},[\cdots[E_{i+1},E_{i}]_{q^{-1}}\cdots]_{q^{-1}},$
$\displaystyle\textrm{ for }(i,j)\in\Delta^{+}.$

We now produce a general coproduct identity that will be used to verify the
tangent space condition for our choice of decomposition of $w_{0}$.

######  Lemma 3.1.

Let $X$ be an element of $U_{q}(\mathfrak{sl}_{n+1})$ whose coproduct admits a
presentation $\Delta(X)=X_{(1)}\otimes X_{(2)}$ such that the second tensor
factor of each summand is an eigenvector of $K_{i}$, with respect to the right
adjoint action of $U_{q}(\mathfrak{sl}_{n+1})$ on itself. Then it holds that

(5)
$\displaystyle\Delta([X,E_{i}]_{q^{-1}})=[X_{(1)},E_{i}]_{\rho_{(2)}-1}\otimes
X_{(2)}K_{i}-X_{(1)}\otimes[X_{(2)},E_{i}]_{q^{-1}},$

where $\rho_{(2)}$ is the eigenvalue of $X_{(2)}$.

###### Proof.

Expanding the expression

$\Delta\left([X,E_{i}]_{q^{-1}}\right)=\Delta(X)\Delta(E_{i})-q^{-1}\Delta(E_{i})\Delta(X)$

we arrive at the sum

$X_{(1)}E_{i}\otimes X_{(2)}K_{i}+X_{(1)}\otimes
X_{(2)}E_{i}-q^{-1}E_{i}X_{(1)}\otimes K_{i}X_{(2)}-q^{-1}X_{(1)}\otimes
E_{i}X_{(2)}.$

Since the second tensor factor of each summand is an eigenvector of $K_{i}$,
this is equal to

$\displaystyle\left(X_{(1)}E_{i}-q^{\rho_{(2)}-1}E_{i}X_{(1)}\right)\\!\otimes
X_{(2)}K_{i}+X_{(1)}\otimes\\!\Big{(}X_{(2)}E_{i}-q^{-1}X_{(2)}E_{i}\Big{)},$

which gives the claimed sum. ∎

######  Example 3.2.

Consider the $q$-deformed commutator $[E_{i},E_{i-1}]_{q^{-1}}$. It follows
from the definition of $U_{q}(\mathfrak{sl}_{n+1})$ that the second tensor
factor of each summand of $\Delta(E_{i})$ is a weight vector. From the formula
above, we now see that $\Delta([E_{i},E_{i-1}]_{q^{-1}})$ is equal to

$\displaystyle[E_{i},E_{i-1}]_{q^{-1}}\otimes
K_{i}K_{i-1}+[1,E_{i-1}]_{q^{-2}}\otimes
E_{i}K_{i-1}+E_{i}\otimes[K_{i},E_{i-1}]_{q^{-1}}+1\otimes[E_{i},E_{i-1}]_{q^{-1}},$

where we have used the fact that $\mathrm{wt}_{i-1}(E_{i})=-1$. This
expression then simplifies to

$\displaystyle[E_{i},E_{i-1}]_{q^{-1}}\otimes K_{i}K_{i-1}+q^{-1}\nu
E_{i-1}\otimes E_{i}K_{i-1}+1\otimes[E_{i},E_{i-1}]_{q^{-1}}.$

We will now use an inductive argument to generalise the formulae given in this
example to the case of $U_{q}(\mathfrak{sl}_{n+1})$.

######  Proposition 3.3.

For any $(i,j)\in\Delta^{+}\\!,$ it holds that

$\displaystyle\Delta(E_{ji})=E_{ji}\otimes K_{j-1,i}\leavevmode\nobreak\
+\leavevmode\nobreak\ q^{-1}\nu\sum_{a=i+1}^{j-1}E_{ai}\otimes
E_{ja}K_{a-1,i}\leavevmode\nobreak\ +\leavevmode\nobreak\ 1\otimes E_{ji},$

where we have denoted $K_{j-1,i}:=K_{j-1}\cdots K_{i}$, for
$(i,j)\in\Delta^{+}\\!,$ and $K_{jj}:=K_{j}$.

###### Proof.

We begin by noting that the claimed identity clearly holds for each simple
generator $E_{i+1,i}=E_{i}$. Let us now assume that the claimed formula holds
for some $E_{ji}$, such that $i\geq 2$. Since the requirements of Lemma 3.1
are satisfied, we can use formula (5) to calculate the coproduct of
$E_{j,i-1}$. The first term of (5) is given by the sum

$\displaystyle[E_{ji},E_{i-1}]_{q^{-1}}\otimes K_{j-1,i-1}\leavevmode\nobreak\
+\leavevmode\nobreak\
q^{-1}\nu\sum_{a=i+1}^{j-1}[E_{ai},E_{i-1}]_{q^{-1}}\otimes E_{ja}K_{a,i-1}$
$\displaystyle+\leavevmode\nobreak\ [1,E_{i-1}]_{q^{-2}}\otimes
E_{ji}K_{i-1},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

which reduces to the sum

$\displaystyle\,E_{j,i-1}\otimes K_{j-1,i-1}\leavevmode\nobreak\
+\leavevmode\nobreak\ q^{-1}\nu\sum_{a=i}^{j-1}E_{a,i-1}\otimes
E_{ja}K_{a,i-1}.$

The second term of (5) is given by the sum

$\displaystyle\,E_{ji}\otimes[K_{j-1,i},E_{i-1}]_{q^{-1}}\leavevmode\nobreak\
+\leavevmode\nobreak\
q^{-1}\nu\sum_{a=i+1}^{j-1}E_{ja}\otimes[E_{ai}K_{j-1,a},E_{i-1}]_{q^{-1}}$
$\displaystyle+\leavevmode\nobreak\
1\otimes[E_{ji},E_{i-1}]_{q^{-1}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

which reduces to the single term $1\otimes E_{j,i-1}$. Adding these two terms
together gives the claimed identity for $E_{j,i-1}$, and hence for the general
case by an inductive argument. ∎

######  Corollary 3.4.

A tangent space is given by

$\displaystyle
T^{(0,1)}=\mathrm{span}_{\mathbb{C}}\\{E_{\beta}\,|\,\beta\in\Delta^{+}\\}.$

###### Proof.

This follows directly from the Proposition 3.3 and the definition of a tangent
space given in §2.2. ∎

Each root vector is a weight vector (see Appendix A), hence
$U_{q}(\mathfrak{h})T^{(0,1)}\subseteq T^{(0,1)}$. The quantum subgroup
associated to $U_{q}(\mathfrak{h})$ is the group Hopf algebra of
$\mathbb{Z}^{n}$, which we denote by $\mathcal{O}(T^{n})$. This implies the
following corollary.

######  Corollary 3.5.

Denote by $(\Omega^{1}_{q}(\mathrm{SU}_{n+1}),\overline{\partial})$ the fodc
associated to the tangent space $T^{(0,1)}$. It holds that
$\Omega^{1}_{q}(\mathrm{SU}_{n+1})$ is right $\mathcal{O}(T^{n})$-covariant.
Moreover, with respect to the associated left $U_{q}(\mathfrak{h})$-action,
the basis elements have weight

$\displaystyle|E_{ji}|=\alpha_{j-1}+\alpha_{j-2}+\cdots+\alpha_{i},$
$\displaystyle\textrm{ for }(i,j)\in\Delta^{+}.$

###  3.2. The Module Structure of the Cotangent Space $\Lambda^{(0,1)}_{q}$

In this subsection we determine the right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module structure of $\Lambda^{(0,1)}_{q}$
by detailing how the matrix generators act on the basis dual to the root
vector basis of $T^{(0,1)}$. We observe that the dual pairing of Hopf algebras
$\left\langle-,-\right\rangle:\mathcal{O}_{q}(\mathrm{SU}_{n+1})\times
U_{q}(\mathfrak{sl}_{n+1})\to\mathbb{C}$, as given in Appendix A, descends to
a non-degenerate pairing

$\displaystyle\left\langle-,-\right\rangle:\Lambda^{(0,1)}_{q}\times
U_{q}(\mathfrak{sl}_{n+1})\to\mathbb{C},$

for which we use the same symbol by abuse of notation.

######  Lemma 3.6.

It holds that

$\displaystyle\langle E_{ji},[u_{rs}]\rangle=\delta_{jr}\delta_{is},$
$\displaystyle\textrm{ for all }(i,j)\in\Delta^{+},\textrm{\emph{ and
}}r,s=1,\dots,n+1.$

Thus a basis of $\Lambda^{(0,1)}_{q}$, dual to the defining basis of
$T^{(0,1)}$, is given by

$\displaystyle\Big{\\{}e_{\alpha_{ji}}:=e_{ji}:=[u_{ji}]\,|\,\textrm{ for
}(i,j)\in\Delta^{+}\Big{\\}}.$

###### Proof.

It follows from the dual pairing formulae given in (39) that the formula holds
for all the simple generators $E_{i}=E_{i+1,i}$. Let us now assume that the
formula holds for some $E_{ji}$. If $i>1$, then it follows from Proposition
3.3 that the pairing

$\displaystyle\langle E_{j,i-1},u_{kl}\rangle=$ $\displaystyle\,\langle
E_{ji}E_{i-1},u_{kl}\rangle-q^{-1}\langle E_{i-1}E_{ji},u_{kl}\rangle$

reduces to the product of Kroenecker delta symbols

$\displaystyle\sum_{a=1}^{n+1}\langle E_{ji},u_{ka}\rangle\langle
E_{i-1},u_{al}\rangle-q^{-1}\sum_{a=1}^{n+1}\langle
E_{i-1},u_{ka}\rangle\langle
E_{ji},u_{al}\rangle=\,\delta_{j+1,k}\delta_{i-1,l}.$

Thus by an inductive argument, the formula holds for all root vectors. ∎

######  Proposition 3.7.

A generating set $G$ for the ideal $I$ associated to the tangent space
$T^{(0,1)}$ is given by the union of the following three sets of elements:

$\displaystyle G_{1}\,:=$
$\displaystyle\\{u_{ij},\,u_{kk}-1,\,|\,(i,j)\in\Delta^{+},k=1,\dots,n+1\\},$
$\displaystyle G_{2}\,:=$
$\displaystyle\\{u_{ji}u_{kl}\,|\,(i,j),\,(k,l)\in\Delta^{+}\\},$
$\displaystyle G_{3}\,:=$
$\displaystyle\\{u_{ji}u_{lk}\,|\,(i,j),\,(k,l)\in\Delta^{+},\,j\neq k\\}.$

Moreover, for $(i,j),\,(j,j^{\prime})\in\Delta^{+}$, and $k=1,\dots,n$, the
actions

(6) $\displaystyle e_{ji}u_{kk}=q^{\delta_{jk}-\delta_{ik}}e_{ji},$
$\displaystyle e_{ji}u_{j^{\prime}j}=\nu e_{j^{\prime}i},$

are the only non-zero actions of the generators $u_{kl}$ on the basis elements
$e_{ji}$.

###### Proof.

It follows from Lemma 3.6 that the elements of $G_{1}$ are contained in the
kernel of $T^{(0,1)}$, which is to say, they are contained in $I$. Consider
now an element $u_{gf}u_{kl}\in G_{2}$, it holds that

$\displaystyle\langle E_{ji},u_{gf}u_{kl}\rangle=$
$\displaystyle\,\langle(E_{ji})_{(1)},u_{gf}\rangle\langle(E_{ji})_{(2)},u_{kl}.\rangle.$

If follows from Proposition 3.3 that $\Delta(E_{ji})$ can be expressed as a
sum of tensor products of root vectors. Now Lemma 3.6 tells us that $u_{kl}$
pairs trivially with any root vector, and so, $u_{gf}u_{kl}$ must be contained
in $I$. Consider next an element $u_{ji}u_{lk}\in G_{3}$, evaluated with a
root vector $E_{ji}$:

$\displaystyle\langle E_{ji},u_{gf}u_{lk}\rangle=$
$\displaystyle\,\langle(E_{ji})_{(1)},u_{gf}\rangle\langle(E_{ji})_{(2)},u_{lk}\rangle.$

It follows from Proposition 3.3 that this expression is equal to

$\displaystyle q^{-1}\nu\sum_{a=i+1}^{j-1}\langle E_{ai},u_{gf}\rangle\langle
E_{ja}K_{a-1,i},u_{lk}\rangle\langle K_{a-1,i},u_{kk}\rangle$

which vanishes since we are assuming that $j\neq k$. Thus it holds that
$G_{3}\subseteq I$.

Let us next show that $G$ generates $I$ as a right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module. Note that an isomorphism of
vector spaces is given by

$\displaystyle\mathcal{O}_{q}(\mathrm{SU}_{n+1})\to\mathcal{O}_{q}(\mathrm{SU}_{n+1})^{+}\oplus\mathbb{C},$
$\displaystyle[a]\mapsto([a^{+}],\varepsilon(a)),$

which restricts to an isomorphism

$\displaystyle\mathcal{O}_{q}(\mathrm{SU}_{n+1})/I\to\Lambda^{(0,1)}_{q}\oplus\mathbb{C}.$

Thus if we can show that

$\dim(\mathcal{O}_{q}(\mathrm{SU}_{n+1})/I)=\dim(\Lambda_{q}(\mathrm{SU}_{n+1}))+1,$

then we can conclude that $G$ is a generating set. Note first that since
$u_{ii}-1\in G$, the coset of any monomial in the diagonal generators is a
scalar multiple of the coset of the identity. Moreover, for any monomial of
the form $fa$, where $f$ is a monomial in the diagonal generators, and
$a\in\mathcal{O}_{q}(\mathrm{SU}_{n+1})$, the coset $[fa]=\varepsilon(f)[a]$.
This means that, if $a=u_{ij}a^{\prime}$, for some $(i,j)\in\Delta^{+}$, and
$a^{\prime}\in\mathcal{O}_{q}(\mathrm{SU}_{n+1})$, then since $u_{ij}\in G$,
we have

$[fu_{ij}a^{\prime}]=\varepsilon(f)[u_{ij}a^{\prime}]=\varepsilon(f)[u_{ji}]a^{\prime}=0.$

Similarly, if $a=u_{ji}a^{\prime}$, then we have

$[fu_{ji}a^{\prime}]=\varepsilon(f)[u_{ji}]a^{\prime}.$

Thus it remains to show that the span of the cosets $[u_{ji}]$, for
$(i,j)\in\Delta^{+}$, is closed under the action of the generators of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. This amounts to showing that every coset
$[u_{ji}u_{ab}]$ is a linear combination of the cosets $[u_{ji}]$. There are
four types of coset to check. First, we have the case

$\displaystyle[u_{ji}u_{kk}]=[u_{kk}u_{ji}]+\mathrm{sign}(j-k)\nu[u_{jk}u_{ki}]=[u_{ji}],$
$\displaystyle\textrm{ for }i>k,\textrm{ or }k>j,$

Secondly, we have the case

$\displaystyle[u_{ji}u_{kk}]=[u_{kk}u_{ji}]=[u_{ji}],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{ for
}j>k>i.$

Finally, we have the case

$\displaystyle[u_{ji}u_{kj}]=[u_{kj}u_{ji}]-\nu[u_{jj}u_{ki}]=-\nu[u_{ki}].$

Moreover, we see that we have also verified the claimed formulae for the
actions of the generators on the basis elements. ∎

###  3.3. Relations of the Maximal Prolongation

In this subsection we calculate the degree two relations for the maximal
prolongation of the fodc $\Omega^{1}_{q}(\mathrm{SU}_{n+1})$, using the
presentation of higher relations given in §2.3. We begin by deriving a general
technical formula.

######  Lemma 3.8.

Let $A$ be a Hopf algebra, endowed with a left $A$-covariant fodc
$\Omega^{1}(A)$. For any $x,y\in A^{+}$, it holds that

$\displaystyle\omega(xy)=[x^{+}y_{(1)}]\otimes[y_{(2)}^{+}]+[y_{(1)}^{+}]\otimes[xy_{(2)}]+[x_{(1)}^{+}y_{(1)}]\otimes[x_{(2)}^{+}y_{(2)}].$

Moreover, for any set of generators $G$ of the right $A$-ideal $I\subseteq
A^{+}$ corresponding to $\Omega^{1}(A)$, it holds that

(7) $\displaystyle
I^{(2)}=\mathrm{span}_{\mathbb{C}}\left\\{\omega(g)a\,|\,\textrm{ for }g\in
G,\,a\in A\right\\}.$

###### Proof.

For any $a,b\in A$, we have that $(ab)^{+}=a^{+}b+\varepsilon(a)b^{+}$. From
this we see that

$\displaystyle\omega(xy)=\,$
$\displaystyle[(x_{(1)}y_{(1)})^{+}]\otimes[(x_{(2)}y_{(2)})^{+}]$
$\displaystyle=\,$
$\displaystyle[x_{(1)}^{+}y_{(1)}+\varepsilon(x_{(1)})y_{(1)}^{+}]\otimes[x_{(2)}^{+}y_{(2)}+\varepsilon(x_{(2)})y_{(2)}^{+}]$
$\displaystyle=\,$
$\displaystyle[x^{+}_{(1)}y_{(1)}]\otimes[x^{+}_{(2)}y_{(2)}]+[xy_{(1)}]\otimes[y_{(2)}^{+}]+[y_{(1)}^{+}]\otimes[xy_{(2)}],$

where we have used the fact that $x,y\in I\subseteq A^{+}$.

We now need to show that, for a set $G$ of generators of the right $A$-ideal
$I$, the elements $\omega(g)$ span the vector space of degree two relations.
Since any other element of $I$ is a linear combination of elements of the form
$ga$, for $g\in G$, $a\in A$, we see that the space of degree two relations is
spanned by elements of the form

$\displaystyle\omega(ga)=[(ga)_{(1)}^{+}]\otimes[(ga)_{(2)}^{+}]=$
$\displaystyle\,[g_{(1)}^{+}a_{(1)}]\otimes[g_{(2)}^{+}a_{(2)}]+[ga_{(1)}]\otimes[a_{(2)}^{+}]+[a_{(1)}^{+}]\otimes[ga_{(2)}]$
$\displaystyle=$
$\displaystyle\,[g_{(1)}^{+}a_{(1)}]\otimes[g_{(2)}^{+}a_{(2)}]+[g]a_{(1)}\otimes[a_{(2)}^{+}]+[a_{(1)}^{+}]\otimes[g]a_{(2)}$
$\displaystyle=$ $\displaystyle\,([g_{(1)}^{+}]\otimes[g_{(2)}^{+}])a$
$\displaystyle=$ $\displaystyle\,\omega(g)a.$

Thus we see that if the space given in (7) is a right $A$-submodule, then it
spans the degree two relations. ∎

Using this lemma, and the set of generators $G=G_{1}\cup G_{2}\cup G_{3}$ from
Lemma 3.7, we will now produce an explicit description of
$\Lambda^{(0,\bullet)}_{q}$ the quantum exterior algebra of
$\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})$.

######  Lemma 3.9.

The algebra $\Lambda^{(0,\bullet)}_{q}$ is isomorphic to the quotient of the
tensor algebra of $\Lambda^{(0,1)}_{q}$ by the right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-ideal generated by the elements,

$\displaystyle e_{ji}\otimes e_{ji^{\prime}}+qe_{ji^{\prime}}\otimes
e_{ji},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ e_{ji}\otimes e_{j^{\prime}i}+qe_{j^{\prime}i}\otimes
e_{ji},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ e_{ji}\otimes
e_{ji}=0,$ $\displaystyle\leavevmode\nobreak\ e_{ji^{\prime}}\otimes
e_{j^{\prime}i}+e_{j^{\prime}i}\otimes e_{ji^{\prime}}+\nu
e_{j^{\prime}i^{\prime}}\otimes e_{ji},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

where $(j,i),(j^{\prime},i^{\prime})\in\Delta^{+}$, and
$j^{\prime}>j>i^{\prime}>i$, and by the elements

$\displaystyle e_{jk}\otimes
e_{j^{\prime}k^{\prime}}+q^{-\delta_{jk^{\prime}}}e_{j^{\prime}k^{\prime}}\otimes
e_{jk},$

where $(j,k),(j^{\prime},k^{\prime})\in\Delta^{+}$, and $j^{\prime}>j$ and
$k^{\prime}>k$.

###### Proof.

It is easily checked that $\omega(g)=0$, for all $g\in G_{1}$. For example,

$\displaystyle\omega(u_{ij})=\sum_{a=1}^{n+1}[u^{+}_{ia}]\otimes[u^{+}_{aj}]=0,$

since $[u_{ij}]=0$, for all $(j,i)\in\Delta^{+}$. For a generator
$u_{ji}u_{kl}\in G_{2}$, it follows from Lemma 3.8 that

$\displaystyle\omega(u_{ji}u_{kl})=\sum_{a=1}^{n+1}[u_{ji}u_{ka}]\otimes[u_{al}]+\sum_{a=1}^{n+1}[u_{ka}]\otimes[u_{ji}u_{al}]+\sum_{a,b=1}^{n+1}[u^{+}_{ja}u_{kb}]\otimes[u^{+}_{ai}u_{bl}]=0,$

where the vanishing of each summand can be concluded from the set of
generators given in Proposition 3.7. Thus we see that the given relations are
generated by the elements of $G_{3}^{+}$.

Let us look at a general element $u_{ji}u_{lk}$ in $G_{3}^{+}$. As we will
see, each generator gives a non-zero degree two relation, whose form depends
on the relationship of the associated pair root vectors $\alpha_{ij}$ and
$\alpha_{kl}$. We group distinct pairs of roots into four types as presented
in the following table.

Type | $(\beta,\gamma)$ |  $\prec$ Comparable
---|---|---
(a) | $>0$ | ✓
(b) | $<0$ | ✓
(c) | $=0$ | ✗
(d) | $=0$ | ✓

Moreover, we find it convenient to consider the situation where the two roots
coincide as an additional Type (e).

Type (a): We further subdivide this case into two sub-cases. First we consider
the situation where $j=l$. We can assume, without loss of generality, that
$k<i$, since $u_{ji}u_{jk}$ is a scalar multiple of $u_{jk}u_{ji}$. In this
case, it holds that

$\displaystyle\omega(u_{ji}u_{jk})=e_{jk}\otimes e_{ji}+qe_{ji}\otimes
e_{jk},$

which gives the first relation.

Next we consider the situation where $i=k$. We can assume, without loss of
generality, that $l<j$, since $u_{ji}u_{li}$ is a scalar multiple of
$u_{li}u_{ji}$. Just as for the previous case, $\omega(u_{ji}u_{li})$ gives
the second relation.

Type (b): Explicitly, for our pair of roots $\alpha_{ji}$ and $\alpha_{lk}$
this is the case for which $i=l$, or of $j=k$. However, since
$u_{ji}u_{lj}\not\in G$, we need only consider the case of $u_{ji}u_{ik}$. We
see

$\displaystyle\omega(u_{ji}u_{ik})=q^{-1}e_{ji}\otimes e_{ik}+e_{ik}\otimes
e_{ji},$

which gives us, up to scalar multiple, the fifth relation for the special case
of $i^{\prime}=j$.

Type (c): Explicitly, this is the case of $j>l$ and $i>k$ or of $l>j$ and
$k>i$. It follows from the relations of $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$
that in the later case

$u_{ji}u_{lk}=u_{lk}u_{ji}+\nu u_{jk}u_{li}.$

Generators of the form $u_{jk}u_{li}$ are consider below in case $(d)$. Hence,
without loss of generality, we can assume that $j>l$ and $i>k$. Making this
assumption, we have

$\omega(u_{ji}u_{lk})=e_{ji}\otimes e_{lk}+e_{lk}\otimes e_{ji},$

which gives us the fifth relation for the special case of $j\neq i^{\prime}$.

Type (d): Explicitly, this is the case of $l<j$ and $k>i$ or of $l>j$ and
$k<i$. Since in both cases the generators $u_{ji}$ and $u_{lk}$ commute, we
can, without loss of generality assume that $j>l$ and $k>i$. This gives us the
relation

$\displaystyle\omega(u_{lk}u_{ji})=e_{lk}\otimes e_{ji}+e_{ji}\otimes
e_{lk}+\nu e_{li}\otimes e_{jk}.$

which implies the fourth relation.

Type (e): Finally we treat the case of $(j,i)=(l,k)$. This gives the relation

$\displaystyle\omega(u_{ji}u_{ji})=\nu e_{ji}\otimes e_{ji},$

which is to say the third relation.

It remains to show that the linear span of this collection of relations is a
right $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-submodule of
$\Lambda^{(0,1)}_{q}\otimes\Lambda^{(0,1)}_{q}$. Observe that any diagonal
generator $u_{kk}$ will act on our relations by a scalar multiple. Moreover,
any generator $u_{kl}$, for $k<l$, will act trivially. Thus we need only look
at the action of strictly lower triangular generators. We treat each of the
five relations on a case by case basis.

1\. The only strictly lower triangular generators that act on relations of the
first type are of the form $u_{pj}$, for some $(j,p)\in\Delta^{+}$.
Multiplying by $u_{pj}$ gives the element

$\displaystyle(e_{ji}\otimes e_{ji^{\prime}}+qe_{ji^{\prime}}\otimes
e_{ji})u_{pj}=$ $\displaystyle\,\nu(e_{ji}\otimes
e_{pi^{\prime}}+qe_{pi}\otimes e_{ji^{\prime}}+qe_{ji^{\prime}}\otimes
e_{pi}+q^{2}e_{pi^{\prime}}\otimes e_{ji})$ $\displaystyle=$
$\displaystyle\,q\nu\left(e_{pi}\otimes e_{ji^{\prime}}+e_{ji^{\prime}}\otimes
e_{pi}+qe_{pi^{\prime}}\otimes e_{ji}+q^{-1}e_{ji}\otimes
e_{pi^{\prime}}\right)$ $\displaystyle=$
$\displaystyle\,q\nu\big{(}e_{pi}\otimes
e_{ji^{\prime}}+e_{ji^{\prime}}\otimes e_{pi}+\nu e_{pi^{\prime}}\otimes
e_{ji}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +q^{-1}(e_{pi^{\prime}}\otimes
e_{ji}+e_{ji}\otimes e_{pi^{\prime}})\big{)}.$

Thus we see that the element lies in the linear span of the given relations.

2\. The strictly lower triangular generators that act on relations of the
second type are of the form $u_{pj}$ or $u_{pj^{\prime}}$, for
$(j,p),(j^{\prime},p)\in\Delta^{+}$. For $u_{pj}$, we see that

$\displaystyle(e_{ji}\otimes e_{j^{\prime}i}+qe_{j^{\prime}i}\otimes
e_{ji})u_{pj}=\nu\Big{(}(1+\theta(p-j^{\prime})q\nu)e_{pi}\otimes
e_{j^{\prime}i}+q^{1+\delta_{pj^{\prime}}}e_{j^{\prime}i}\otimes
e_{pi}\Big{)},$

where $\theta$ is the Heaviside step function. This needs to be further
divided into the following three cases, depending on whether $p<j^{\prime}$,
$p=j^{\prime}$, or $p>j^{\prime}$. It is then easily checked that for each
case we again get a scalar multiple of one of the given relations. The case of
the generator $u_{pj^{\prime}}$ is dealt with similarly.

3\. The only non-diagonal generators that act on the relation of the third
type are of the form $u_{pj}$, for $(j,p)\in\Delta^{+}$. For such a generator,
we see that

$\displaystyle(e_{ji}\otimes e_{ji})u_{pj}=\nu(e_{ji}\otimes
e_{pi}+qe_{pi}\otimes e_{ji}).$

Thus we get a scalar multiple of a relation of the second type.

4\. The only strictly lower triangular generators that act on relations of the
fourth type are of the form $u_{pj}$ or $u_{pj^{\prime}}$, for
$(p,j),\,(p,j^{\prime})\in\Delta^{+}$. Multiplying the relation
$e_{j^{\prime}i^{\prime}}\otimes e_{ji}+q^{\delta_{i^{\prime}j}}e_{ji}\otimes
e_{j^{\prime}i^{\prime}}$ by the generator $u_{pj}$, we get the expression

$\displaystyle
q^{\delta_{i^{\prime}j}}\nu\\!\left(q^{\delta_{j^{\prime}p}-\delta_{i^{\prime}p}}e_{j^{\prime}i^{\prime}}\otimes
e_{pi}+e_{pi}\otimes e_{j^{\prime}i^{\prime}}+\theta(p-j^{\prime})\nu
e_{pi^{\prime}}\otimes e_{j^{\prime}i}\right)\\!.$

This needs to be further divided into the following three cases, depending on
whether $p<j^{\prime}$, $p=j^{\prime}$, or $p>j^{\prime}$. It is then easily
checked that for each case we again get a scalar multiple of one of the given
relations. The simpler case of the generator $u_{pj^{\prime}}$ is dealt with
similarly.

5\. A relation of the fifth type can be acted on non-trivially by strictly
lower triangular generators of the form $u_{pj}$ or $u_{pj^{\prime}}$, for
$(j,p),(j^{\prime},p)\in\Delta^{+}$. Multiplying by $u_{pj}$, and dividing by
$\nu$, gives the element

$\displaystyle e_{pi^{\prime}}\otimes
e_{j^{\prime}i}+q^{\delta_{pj^{\prime}}}(e_{j^{\prime}i}\otimes
e_{pi^{\prime}}+\nu e_{j^{\prime}i^{\prime}}\otimes
e_{pi})+\theta(p-j^{\prime})\nu\left(e_{pi}\otimes
e_{j^{\prime}i^{\prime}}+\nu e_{pi^{\prime}}\otimes e_{j^{\prime}i}\right).$

This needs to be further divided into the following three cases, depending on
whether $p<j^{\prime}$, $p=j^{\prime}$, or $p>j^{\prime}$. It is then easily
checked that for each case we again get a linear combination of the given
relations. The simpler case of the generator $u_{pj^{\prime}}$ is dealt with
similarly.

From this we can conclude that the linear span of the set
$\\{\omega(g)\,|\,\textrm{ for }g\in G\\}$ is closed under the right action of
$\mathcal{O}_{q}(\mathrm{SU}_{n})$. It now follows from Lemma 3.8 that the
given relations span the space of degree two relations. ∎

We will now produce a compact presentation of the relations, given in terms of
the $A_{n}$-root system. Interestingly, the pairs of basis elements that fail
to commute up to a scalar correspond to pairs of positive roots which are
comparable with respect to the partial order $\prec$ on $\Delta$. For the
statement of the result, we find it convenient to introduce the notion of the
_prime pair_ of a pair of orthogonal roots
$\\{\beta,\gamma\\}=\\{\alpha_{ab},\alpha_{cd}\\}$:

$\\{\beta^{\prime},\gamma^{\prime}\\}=\\{\alpha_{ab}^{\prime},\alpha_{cd}^{\prime}\\}:=\\{\alpha_{ad},\alpha_{cb}\\}.$

It is important to note that if $\beta$ and $\gamma$ are comparable, and both
are positive, then $\beta^{\prime}$ and $\gamma^{\prime}$ are also both
positive. We will also use the convex order $\leq$ associated to our choice of
reduced decomposition of $w_{0}$, as presented in Appendix A.1. In particular,
we put an ordering on the basis elements of the cotangent space, by setting

$\displaystyle e_{\gamma}<e_{\gamma^{\prime}},$ $\displaystyle\text{ if
}\gamma<\gamma^{\prime},$

with respect to the convex order on $\Delta^{+}$.

######  Theorem 3.10.

For all $\beta\leq\gamma\in\Delta^{+}$, the relations

$\displaystyle e_{\beta}\wedge e_{\gamma}\,=$
$\displaystyle\,-e_{\gamma}\wedge e_{\beta}+\nu\,e_{\beta^{\prime}}\wedge
e_{\gamma^{\prime}},$ $\displaystyle\textrm{ for }(\beta,\gamma)=0\textrm{ and
}\beta\prec\gamma,$ $\displaystyle e_{\beta}\wedge e_{\gamma}\,=$
$\displaystyle\,-q^{(\beta,\gamma)}e_{\gamma}\wedge e_{\beta},$
$\displaystyle\textrm{ otherwise},$

give a full set of relations for the algebra $\Lambda^{(0,\bullet)}_{q}$.

###### Proof.

We establish this presentation by showing that, for each pair of positive
roots $\\{\beta,\gamma\\}$, the relation given above coincides with a
corresponding relation in Lemma 3.9, and that all relations in Lemma 3.9 arise
in this way. Note first that a pair of positive orthogonal roots
$\\{\beta,\gamma\\}$ satisfies $\beta\prec\gamma$ if and only if it is of the
form $\\{\alpha_{ij^{\prime}},\alpha_{i^{\prime}j}\\}$, for
$i<i^{\prime},\,j<j^{\prime}$. Thus, for this case, the relation given above
corresponds to the fourth relation in Lemma 3.9. The case when $\beta$ and
$\gamma$ are positive, orthogonal, and _not_ comparable happens precisely when
the pair of roots is of the form
$\\{\alpha_{kj},\alpha_{k^{\prime}j^{\prime}}\\}$, for $j^{\prime}\neq k$, and
$k<k^{\prime},\,j<j^{\prime}$. Thus this case corresponds to the fifth
relation in Lemma 3.9, for $j^{\prime}\neq k$.

We have $\langle\beta,\gamma\rangle<0$ precisely when the pair of roots is of
the form $\\{\alpha_{jk},\alpha_{kk^{\prime}}\\}$, for $k<k^{\prime}$. Thus
the relation above corresponds to the fifth relation in Lemma 3.9, for
$j^{\prime}=k$. Next we consider the case when $(\beta,\gamma)>0$. This
happens precisely when the pair of roots is of the form
$\\{\alpha_{ij},\alpha_{ij^{\prime}}\\}$, for $j<j^{\prime}$, or
$\\{\alpha_{ij},\alpha_{i^{\prime}j}\\}$, for $i<i^{\prime}$. For both pairs
of roots $(\beta,\gamma)=1$ meaning that we recover the first and second
relations in Lemma 3.9. Finally, the case $\beta=\gamma$ corresponds to the
third relation in Lemma 3.9. ∎

###  3.4. Some Low Rank Examples

In this subsection, we find it instructive to present some low rank examples.
We begin with the simplest case of $U_{q}(\mathfrak{sl}_{2})$. We then move to
the case of $U_{q}(\mathfrak{sl}_{3})$, whose structure, while still quite
simple, exhibits much more interesting behaviour. Finally, we look at the case
of $U_{q}(\mathfrak{sl}_{4})$ which exhibits behaviour more characteristic of
the general case.

######  Example 3.11.

For the simplest case of $\mathcal{O}_{q}(\mathrm{SU}_{2})$, we see that
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{2})$ is the one-dimensional vector space
spanned by the basis element $e:=e_{21}=[u_{21}]$, and its right
$\mathcal{O}_{q}(\mathrm{SU}_{2})$-module structure is determined by
$eu_{11}=q^{-1}e_{11}$, and $eu_{22}=qe_{22}$, with all other generators
$u_{ij}$ acting trivially. Since $e\wedge e=0$, we have
$\Lambda^{k}_{q}(\mathrm{SU}_{2})=0$, for all $k>1$.

######  Example 3.12.

The case of $\mathcal{O}_{q}(\mathrm{SU}_{3})$ is more instructive. Here the
cotangent space $\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})$ is a three-dimensional
vector space with basis

$\displaystyle e_{21}=[u_{21}]\leavevmode\nobreak\ $ $\displaystyle
e_{31}=[u_{31}]\leavevmode\nobreak\ $ $\displaystyle
e_{32}=[u_{32}].\leavevmode\nobreak\ $

The diagonal matrix generators of $\mathcal{O}_{q}(\mathrm{SU}_{3})$ act on
the basis elements by a power of $q$, just as for
$\mathcal{O}_{q}(\mathrm{SU}_{2})$. Moreover, we have a single non-diagonal
action

$\displaystyle e_{21}u_{32}=\nu e_{31}.$

In what follows, we find it instructive to present the non-diagonal actions in
the form of a graph: Arrange the basis elements in lower triangular form and
draw an arrow from one basis element $e$ to another $e^{\prime}$ if there
exists a generator $u_{ji}$ such that $eu_{ji}=\nu e^{\prime}$. Thus the graph
for $U_{q}(\mathfrak{sl}_{3})$ is as follows:

The relations of $\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})$ can be conveniently
presented in terms of the following diagram:

where the dotted arrow highlights the fact that it points in the opposite
direct to our choice of convex order. This gives the relations

$\displaystyle e_{\beta}\wedge e_{\beta}=0,$ $\displaystyle e_{\beta}\wedge
e_{\gamma}=-qe_{\gamma}\wedge e_{\beta},$

whenever an arrow points from a basis element $e_{\beta}$ to a basis element
$e_{\gamma}$. Note that the reversal of the dotted arrow follows from the fact
that the inner product of the two roots $\alpha_{12}$ and $\alpha_{23}$ is
negative.

Thus we see that $\Lambda^{(0,\bullet)}_{q}(\mathrm{SU}_{3})$ is a quantum
affine space in the sense of [3, I.2], and that the higher orders have
classical binomial dimension:

$\displaystyle\dim\\!\left(\Lambda^{(0,2)}_{q}(\mathrm{SU}_{3})\right)=3,$
$\displaystyle\dim\\!\left(\Lambda^{(0,3)}_{q}(\mathrm{SU}_{3})\right)=1,$

with all higher forms being zero.

That the commutation relations take such a simple form can be understood as
following from the explicit dimensions of the weight spaces of
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})\otimes\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})$:

$\displaystyle\begin{pmatrix}\,|e_{1}\otimes e_{1}|&|e_{1}\otimes
e_{2}|&|e_{1}\otimes e_{3}|\,\\\ \,|e_{2}\otimes e_{1}|&|e_{2}\otimes
e_{2}|&|e_{2}\otimes e_{3}|\,\\\ \,|e_{3}\otimes e_{1}|&|e_{3}\otimes
e_{2}|&|e_{3}\otimes e_{3}|\,\\\ \end{pmatrix}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\begin{pmatrix}2\alpha_{1}&\alpha_{1}+\alpha_{2}&2\alpha_{1}+\alpha_{2}\\\
\alpha_{1}+\alpha_{2}&2\alpha_{2}&\alpha_{1}+2\alpha_{2}\\\
\,2\alpha_{1}+\alpha_{2}&\alpha_{1}+2\alpha_{2}&2\alpha_{3}\\\
\end{pmatrix}_{.}$

Since the calculus is right $\mathcal{O}(T^{2})$-covariant, it follows that
$I^{(2)}$ is a $U_{q}(\mathfrak{h})$ submodule of
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})\otimes\Lambda^{(0,1)}_{q}(\mathrm{SU}_{3})$.
In other words, $I^{(2)}$ is homogeneous with respect to the
$\mathbb{Z}^{2}$-grading of the tensor product. Thus we can choose a basis of
the degree two relations which is of the form

$\displaystyle e_{ji}\otimes e_{kl}+\lambda e_{kl}\otimes e_{ji},$
$\displaystyle e_{ji}\otimes e_{ji},$ $\displaystyle\textrm{ where
}(j,i),(k,l)\in\Delta^{+}.$

Thus we see that, for this simple case, the form of the relations is dictated
by the dimension of the weight spaces.

We finish with an observation that the commutation relations of the root
vectors $E_{ji}$ are intimately connected with the commutation relations of
the dual forms $e_{ji}$. For sake of notational convenience, we first relabel
the root vectors

$\displaystyle X_{1}:=E_{21}=E_{1},$ $\displaystyle
X_{2}:=E_{31}:=[E_{2},E_{1}]_{q^{-1}},$ $\displaystyle X_{3}:=E_{32}=E_{2}.$

For any $x\in I$, we see that

$\displaystyle[x_{(1)}]\otimes[x_{(2)}]=$ $\displaystyle\,\sum_{i,j}\langle
X_{i},x_{(1)}\rangle\langle X_{j},x_{(2)}\rangle e_{i}\otimes
e_{j}=\,\sum_{i,j=1}^{3}\langle X_{i}X_{j},x\rangle e_{i}\otimes e_{j}.$

Separating this into its homogeneous summands we see that the elements

(8) $\displaystyle\langle X_{i}X_{j},x\rangle e_{i}\otimes e_{j}+\langle
X_{j}X_{i},x\rangle e_{j}\otimes e_{i},$ $\displaystyle\langle
X_{i}^{2},x\rangle e_{i}\otimes e_{i},$ $\displaystyle\text{
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ for }1\leq
i<j\leq 3,$

are all contained in $I^{(2)}$. Now in the linear dual of the ideal $I$, we
have the following $q$-commutation relations

$\displaystyle X_{1}X_{2}=q^{-1}X_{2}X_{1},$ $\displaystyle
X_{1}X_{3}=qX_{3}X_{1},$ $\displaystyle X_{2}X_{3}=q^{-1}X_{3}X_{2},$

where the first and third relations follow from the quantum Serre relations of
$U_{q}(\mathfrak{sl}_{3})$. This allows us to rewrite (8) as

$\displaystyle\langle E_{i}^{2},x\rangle e_{i},$ $\displaystyle\langle
X_{i}X_{j},x\rangle(e_{i}\otimes e_{j}+q^{\chi_{ij}}e_{j}\otimes
e_{I}),\textrm{ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
for }1\leq i<j\leq 3,$

where $\chi_{12}=\chi_{23}=1$ and $\chi_{13}=-1$. Thus we see that the
dimension of the space of relations is bounded above by six. This means that
$\omega(G)$ (where $G$ is the generating set given in Proposition 3.7) and
$I^{(2)}$ coincide. This offers an alternative proof for the fact that
$\omega(G)$ is closed under the right action of
$\mathcal{O}_{q}(\mathrm{SU}_{3})$, as established in Lemma 3.9.

######  Example 3.13.

Finally, we treat the case of $U_{q}(\mathfrak{sl}_{4})$. The $6$-dimensional
cotangent space $\Lambda^{(0,1)}_{q}(\mathrm{SU}_{4})$ has four non-diagonal
matrix generator actions $u_{ji}$ which we present in the form of a diagram,
just as for $U_{q}(\mathfrak{sl}_{3})$:

We note that as an $\mathcal{O}_{q}(\mathrm{SU}_{4})$-module
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{4})$ decomposes into a sum of three obvious
submodules. Moreover, the first two submodules are reducible, and consist of a
series of one-dimensional extensions of a one-dimensional submodule.

In this case the algebra $\Lambda^{(0,\bullet)}_{q}(\mathrm{SU}_{4})$ is no
longer a quantum affine space. This can be seen as a consequence of the fact
that the weight space of weight $\alpha_{1}+2\alpha_{2}+\alpha_{3}$ in
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{4})\otimes\Lambda^{(0,1)}_{q}(\mathrm{SU}_{4})$
is four dimensional. This permits the relation

(9) $\displaystyle e_{32}\wedge e_{41}=-e_{41}\wedge e_{32}+\nu e_{31}\wedge
e_{42}.$

Let us now give a graphic presentation of the relations, just as we did for
$U_{q}(\mathfrak{sl}_{3})$ above. For the case when the roots have a non-zero
pairing we have the following diagrams

where for any two basis elements $e_{\beta}$ and $e_{\gamma}$ connected by a
red line, such that $\beta\leq\gamma$, we have the relation

$e_{\beta}\wedge e_{\gamma}=-qe_{\gamma}\wedge e_{\beta}.$

When the pairing of roots is zero, we have the incomparable and comparable
situations:

In the incomparable case all pairs skew commute, while for the comparable case
we have the relation (9), where the dotted line in the diagram joins the two
factors of the extra _quantum term_ appearing on the right hand side of (9).

###  3.5. A Filtration for the Quantum Exterior Algebra

In this subsection, we introduce a filtration on the quantum exterior algebra
and describe its associated graded algebra. Just as for the irreducible case
treated in [22], the filtration will be used in the next subsection to show
that our quantum exterior algebra is a Frobenius algebra.

We begin by noting that our choice of basis for the cotangent space
$\Lambda^{(0,1)}_{q}$ is indexed by the set $\Delta^{+}$ of positive roots of
$\mathfrak{sl}_{n+1}$. Moreover, our choice of reduced decomposition of the
longest element of the Weyl group gives $\Delta^{+}$ a total order (as
explained in Appendix A.1). Thus, following the approach of Appendix B, we
have an associated $\mathcal{Q}^{+}$-monoid grading on the tensor algebra of
$\Lambda^{(0,1)}_{q}$. Finally, this induces a filtration on the quantum
exterior algebra. In the following lemma, we use Proposition B.1 to describe
the associated graded algebra $\mathrm{gr}^{\mathscr{F}}$.

######  Lemma 3.14.

The associated graded algebra of the filtration is generated by the elements
$e_{\gamma}$, for $\gamma\in\Delta^{+}$, subject to the relations

(10) $\displaystyle e_{\beta}\otimes e_{\beta},$ $\displaystyle
e_{\beta}\otimes e_{\gamma}=-q^{(\beta,\gamma)}e_{\gamma}\otimes e_{\beta},$
$\displaystyle\textrm{ for }\beta<\gamma\in\Delta^{+}.$

###### Proof.

Proposition B.1 tells us that the leading terms of the relation set for the
quantum exterior algebra give a relation set for the associated graded
algebra. For $(\beta,\gamma)=0$ and $\beta\prec\gamma$, we have the relation

$e_{\beta}\otimes e_{\gamma}+e_{\gamma}\otimes
e_{\beta}-\nu\,e_{\beta^{\prime}}\otimes e_{\gamma^{\prime}}.$

We see that the first two terms have the degree $\beta+\gamma$ and the third
term has degree $\beta^{\prime}+\gamma^{\prime}$. Since
$\beta+\gamma<\beta^{\prime}+\gamma^{\prime}$ in $\mathcal{Q}^{+}$, the
leading term is

$e_{\beta}\otimes e_{\gamma}+e_{\gamma}\otimes e_{\beta}=e_{\beta}\otimes
e_{\gamma}+q^{(\beta,\gamma)}e_{\gamma}\otimes e_{\beta}.$

For two distinct roots $\beta,\gamma$ whose pairing is zero, or which are not
comparable, we have the relation

$e_{\beta}\otimes e_{\gamma}+q^{(\beta,\gamma)}e_{\gamma}\otimes e_{\beta}.$

Since both summands have degree $\beta+\gamma$, the relation is equal to its
leading term. Finally, it is clear that $e_{\beta}\otimes e_{\beta}$ is equal
to its leading term. Thus we have a full set of relations. ∎

Thus we see that the associated graded algebra is a quantum affine algebra,
and that it is a $2^{|\Delta^{+}|}$ – dimensional vector space. We will now
use this information to produce a basis for our quantum exterior algebra.

######  Proposition 3.15.

For $k=1,\dots,|\Delta^{+}|$, a basis of $\Lambda_{q}^{(0,k)}$ is given by

$\displaystyle\Big{\\{}e_{\gamma_{1}}\wedge\cdots\wedge
e_{\gamma_{k}}\,|\,\gamma_{1}<\cdots<\gamma_{k}\in\Delta^{+}\Big{\\}}.$

In particular, it holds that

$\displaystyle\mathrm{dim}\Big{(}V^{(0,k)}\Big{)}=\binom{\,|\Delta^{+}|\,}{k},$
and $\displaystyle\mathrm{dim}\Big{(}V^{(0,\bullet)}\Big{)}=2^{|\Delta^{+}|}.$

###### Proof.

It is clear from the relation set given in Theorem 3.10 that the proposed
basis is a spanning set. Just as for any finite-dimensional filtered algebra,
the dimension of the graded algebra $\mathrm{gr}^{\mathscr{F}}$ must be equal
to the dimension of original filtered algebra $\Lambda^{(0,\bullet)}_{q}$,
which is to say $2^{|\Delta^{+}|}$, and so, the proposed set is a basis. ∎

######  Example 3.16.

For the special case of $U_{q}(\mathfrak{sl}_{3})$, the order on the cotangent
basis elements is given by

$\displaystyle e_{(2,1)}\leavevmode\nobreak\ <\leavevmode\nobreak\
e_{(3,1)}\leavevmode\nobreak\ <\leavevmode\nobreak\ e_{(3,2)}.$

Thus for tensor products of order two, we see for example that

$\displaystyle e_{(2,1)}\otimes e_{(3,1)}\leavevmode\nobreak\
<\leavevmode\nobreak\ e_{(3,2)}\otimes e_{(2,1)}\leavevmode\nobreak\
<\leavevmode\nobreak\ e_{(3,1)}\otimes e_{(3,2)}.$

Moreover, since all the relations are clearly equal to their leading terms, we
see that the associated graded algebra $\mathrm{gr}^{\mathscr{F}}$ is
isomorphic to $\Lambda^{(0,\bullet)}_{q}$.

######  Example 3.17.

Let us now look at the more involved case of $U_{q}(\mathfrak{sl}_{4})$. An
explicit presentation of the lexicographical order of the basis elements of
the cotangent space is given by the diagram

where the total order is the transitive closure of the relations implied by
the arrows of the diagram. Consider now the single non-$q$-skew commutation
relation

$e_{(3,2)}\otimes e_{(4,1)}+e_{(4,1)}\otimes e_{(3,2)}+\nu e_{(4,2)}\otimes
e_{(3,1)}.$

Since $e_{(3,1)}<e_{(3,2)}$, we see that the leading term is given by

$\displaystyle e_{(3,2)}\otimes e_{(4,1)}+e_{(4,1)}\otimes e_{(3,2)}$

showing that the associated graded algebra is a quantum affine algebra.

###  3.6. The Frobenius and Koszul Properties

In this subsection we establish some important additional algebraic properties
of the quantum exterior algebra. We use the associated graded algebra
$\mathrm{gr}^{\mathscr{F}}$ to prove that the quantum exterior algebra is a
Frobenius algebra, and then explicitly describe its Nakayama automorphism.
Moreover, we observe that the quantum exterior algebra is a Koszul algebra.

A _Frobenius algebra_ is an associative algebra $A$ equipped with a non-
degenerate bilinear map $B:A\times A\to\mathbb{C}$ satisfying

(11) $\displaystyle B(ab,c)=B(a,bc),$ $\displaystyle\textrm{ for all }a,b,c\in
A.$

######  Proposition 3.18.

For any linear isomorphism
$\iota:\mathrm{gr}^{\mathscr{F}}_{|\Delta^{+}|}\to\mathbb{C}$, the bilinear
map

$\displaystyle
B:\mathrm{gr}^{\mathscr{F}}\otimes\mathrm{gr}^{\mathscr{F}}\to\mathbb{C},$
$\displaystyle v\otimes w\mapsto\iota(v\wedge w)$

gives $\mathrm{gr}^{\mathscr{F}}$ the structure of a Frobenius algebra.

###### Proof.

It follows from the relation set given in (10) that, for any
$v\in\mathrm{gr}^{\mathscr{F}}$, there exists a $v^{\prime}$ such that

$\displaystyle v\wedge
v^{\prime}=\lambda\,\prod_{\gamma\in\Delta^{+}}e_{\gamma},$
$\displaystyle\textrm{ for some }\lambda\in\mathbb{C}.$

Thus $B$ is a non-degenerate bilinear form. Since (11) is clearly satisfied,
we see that $B$ gives $\mathrm{gr}^{\mathscr{F}}$ the structure of a Frobenius
algebra. ∎

The corollary below now follows from Bongale’s theorem for filtered algebras,
as presented in Appendix B.

######  Corollary 3.19.

For any linear isomorphism
$\iota:\Lambda^{(0,|\Delta^{+}|)}_{q}\to\mathbb{C}$, the bilinear map

$\displaystyle
B:\Lambda^{(0,\bullet)}_{q}\otimes\Lambda^{(0,\bullet)}_{q}\to\mathbb{C},$
$\displaystyle v\otimes w\mapsto\iota(v\wedge w)$

gives $\mathrm{gr}^{\mathscr{F}}$ the structure of a Frobenius algebra.

###### Proof.

From Bongale’s theorem we know that a Frobenius structure is given by

$\displaystyle
B:\Lambda^{(0,\bullet)}_{q}\otimes\Lambda^{(0,\bullet)}_{q}\to\mathbb{C},$
$\displaystyle v\otimes w\mapsto\iota([v\wedge w]_{|\Delta^{+}|}),$

where $\iota:\mathrm{gr}^{\mathscr{F}}_{2\rho}\simeq\mathbb{C}$ is some choice
of linear isomorphism, and $\rho$ is the half-sum of positive roots of
$\mathfrak{sl}_{n+1}$. The given $B$ now follows from the fact that
$\mathrm{gr}^{\mathscr{F}}_{2\rho}$ and $\Lambda^{(0,|\Delta^{+}|)}_{q}$ are
obviously linearly isomorphic. ∎

For a general Frobenius algebra $A$, there exists an algebra automorphism
$\sigma:A\to A$ of $A$, uniquely defined by the identity
$B(x,y)=B(y,\sigma(x))$, for all $x,y\in A$. We see that the bilinear form $B$
of a Frobenius algebra is symmetric if and only if $\sigma=\mathrm{id}$. With
a view to describing the Nakayama automorphism of our quantum exterior
algebra, we first describe the Nakayama automorphism of
$\mathrm{gr}^{\mathscr{F}}$.

######  Proposition 3.20.

The Nakayama automorphism of $\mathrm{gr}^{\mathscr{F}}$ is determined by

$\displaystyle\sigma(e_{\gamma})=(-1)^{|\Delta^{+}|-1}e_{\gamma},$
$\displaystyle\textrm{ for any }\gamma\in\Delta^{+}.$

###### Proof.

Note first that, for any $\gamma\in\Delta^{+}$, the only basis element that
pairs non-trivially with $e_{\gamma}$ is the element

$\prod_{\leavevmode\nobreak\ \gamma\,\neq\,\theta\,\in\Delta^{+}}e_{\theta}.$

It follows from the relations given in Theorem 3.10 that

(12) $\displaystyle
B\\!\left(e_{\gamma},\prod_{\gamma\,\neq\,\theta\,\in\Delta^{+}}e_{\theta}\right)=(1)^{|\Delta^{+}|-1}q^{\chi_{\gamma}}B\left(\prod_{\gamma\,\neq\,\theta\,\in\Delta^{+}}e_{\theta},e_{\gamma}\right)\\!,$

where we have denoted

$\chi_{\gamma}:=-\sum_{\gamma>\theta\in\Delta^{+}}\,(\gamma,\theta)+\sum_{\gamma<\theta\in\Delta^{+}}\,(\gamma,\theta).$

Now for an arbitrary positive root $\alpha_{ij}\in\Delta^{+}$, the positive
roots that pair non-trivially with $\alpha_{ij}$ are precisely those of the
form

$\displaystyle\alpha_{ik},\leavevmode\nobreak\
\alpha_{fj},\leavevmode\nobreak\ \alpha_{ai},\leavevmode\nobreak\
\alpha_{jb},$ $\displaystyle\textrm{ for
}(i,k),(f,j),(a,i),(j,b)\in\Delta^{+}.$

Let us now collect together those roots that are strictly less then
$\alpha_{ij}$ with respect to the convex order $\leq$. We then pair them with
$\alpha_{ij}$ and sum the resulting scalars to give

$\displaystyle\sum_{k=i+1,}^{j-1}(\alpha_{ij},\alpha_{ik})+\sum_{f=1}^{i-1}(\alpha_{ij},\alpha_{fj})+\sum_{a=1}^{i-1}(\alpha_{ij},\alpha_{ai})=j-i-1,$

where we have observed that the last two summands cancel. Following the same
procedure for those roots that are strictly greater then $\alpha_{ij}$, we get

$\displaystyle\sum_{k=j+1}^{n+1}(\alpha_{ij},\alpha_{ik})+\sum_{f=i+1}^{j-1}(\alpha_{ij},\alpha_{fj})+\sum_{b=j+1}^{n+1}(\alpha_{ij},\alpha_{jb})=j-i-1,$

where we have observed that the first and the last summands cancel. Thus

$\displaystyle\sum_{\gamma<\theta\in\Delta^{+}}(\gamma,\beta)=\sum_{\gamma>\theta\in\Delta^{+}}(\gamma,\beta),$

meaning that the identity in (12) reduces to

(13) $\displaystyle
B\\!\left(e_{\gamma},\prod_{\gamma\neq\theta\in\Delta^{+}}e_{\theta}\right)=(-1)^{|\Delta^{+}|-1}B\\!\left(\prod_{\gamma\neq\theta\in\Delta^{+}}e_{\theta},e_{\gamma}\right)\\!.$

This identifies the Nakayama automorphism on the algebra generators
$e_{\gamma}$, and hence on the whole algebra $\mathrm{gr}^{\mathscr{F}}$. ∎

In the following corollary we use our description of the Nakayama automorphism
of $\mathrm{gr}^{\mathscr{F}}$ to produce an analogous description of the
Nakayama automorphism of $\Lambda^{(0,\bullet)}_{q}$.

######  Corollary 3.21.

The Nakayma automorphism of $\Lambda^{(0,\bullet)}_{q}$ is determined by

$\displaystyle\sigma(e_{\gamma})=(-1)^{|\Delta^{+}|-1}e_{\gamma},$
$\displaystyle\textrm{ for any }\gamma\in\Delta^{+}.$

###### Proof.

For any $\gamma\in\Delta^{+}$, let $v$ be an element of
$\Lambda^{(0,\,|\Delta^{+}|-1)}_{q}$ satisfying

$0\neq e_{\gamma}\wedge v\in\Lambda^{(0,\,|\Delta^{+}|)}_{q}.$

Since $\Lambda^{(0,\,|\Delta^{+}|)}_{q}$ is a one-dimensional space, there
exists a scalar $c$ such that $e_{\gamma}\wedge v=cv\wedge e_{\gamma}.$ Acting
on this product by the map $f$ (as introduced in Appendix B) we get the
product $f(e_{\gamma})\wedge f(v)=cf(v)\wedge f(e_{\gamma}).$ (Note that, by
abuse of notation, we have not differentiated notationally between the product
of $\Lambda^{(0,\bullet)}_{q}$ and the product of
$\mathrm{gr}^{\mathscr{F}}$.) However, we already know from the presentation
of the Nakayama automorphism of $\mathrm{gr}^{\mathscr{F}}$ that

$f(e_{\gamma})\wedge v=(-1)^{|\Delta^{+}|-1}v\wedge f(e_{\gamma}).$

Thus we must conclude that $c=(-1)^{|\Delta^{+}|-1}$, confirming the proposed
presentation of the Nakayama automorphism. ∎

Thus we see that, just as in the classical case, the Frobenius structure of
$\mathrm{gr}^{\mathscr{F}}$ and $\Lambda^{(0,\bullet)}_{q}$ is _graded
symmetric_ , which is to say, for any two homogeneous elements $v$ and $w$ we
have

$B(v,w)=(-1)^{|v||w|}B(w,v).$

Indeed, when $|\Delta^{+}|$ is odd, both Frobenius algebras are symmetric.

We finish by observing that $\Lambda^{(0,\bullet)}_{q}$ is also a Koszul
algebra. Recall that a Koszul algebra is a $\mathbb{Z}_{\geq 0}$-graded
algebra admitting a linear minimal graded free resolution. We refer the reader
to the standard text [35] for more details on Koszul algebras.

######  Proposition 3.22.

The algebra $\Lambda^{(0,\bullet)}_{q}$ is a Koszul algebra.

###### Proof.

The algebra $\Lambda^{(0,\bullet)}_{q}$ is clearly a PBW-algebra in the sense
of Priddy [35, §4.1]. Thus it follows from Priddy’s theorem [35, Theorem 3.1]
that it is Koszul. ∎

##  4\. The Heckenberger–Kolb Dolbeault Complex for the Quantum Grassmannians

In this section we examine the restriction of the dc
$\Omega_{q}^{(0,\bullet)}(\mathrm{SU}_{n+1})$ to the quantum Grassmannians and
show that we recover the anti-holomorphic Heckenberger–Kolb dc. The embedding
of the higher Heckenberger–Kolb forms into the higher
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-forms is a delicate issue, and
necessitates the introduction of some general results in §4.5.

###  4.1. Some Preliminaries on Quantum Homogeneous Spaces

In this preliminary subsection, we recall the definition of a quantum
homogeneous space, and the associated generalisations of the fundamental
theorem of Hopf modules.

Let $A$ be a Hopf algebra. We say that a left coideal subalgebra $B\subseteq
A$ is a _quantum homogeneous $A$-space_ if $A$ is faithfully flat as a right
$B$-module and $B^{+}A=AB^{+}$. In [44, Theorem 1] it was shown that, for the
Hopf algebra surjection $\pi_{B}:A\to A/B^{+}A$, with its associated right
$\pi_{B}(A)$-coaction
$\Delta_{R,\pi_{B}}:=(\mathrm{id}\otimes\pi_{B})\circ\Delta:A\to
A\otimes\pi_{B}(A),$ the space of coinvariants

$A^{\mathrm{co}(\pi_{B}(A))}:=\Big{\\{}b\in
A\,|\,\Delta_{R,\pi_{B}}(b)=b\otimes 1\Big{\\}}$

is equal to $B$. We call $\pi_{B}(A)$ the _isotropy Hopf algebra_ of $B$.

For any quantum homogeneous space $B=A^{\mathrm{co}(\pi_{B}(A))}$, we define
${}^{A}_{B}\mathrm{Mod}_{B}$ to be the category whose objects are left
$A$-comodules $\Delta_{L}:\mathcal{F}\to A\otimes\mathcal{F}$, endowed with a
$B$-bimodule structure such that
$\Delta_{L}(bfc)=\Delta(b)\Delta_{L}(f)\Delta(c)$, for all
$f\in\mathcal{F},\,b,c\in B$, and whose morphisms are left $A$-comodule,
$B$-bimodule, maps. We call this category the _category of $(A,B)$-relative
Hopf modules_.

Let ${}^{\pi_{B}\,}\\!\mathrm{Mod}_{B}$ denote the category whose objects are
given by left $\pi_{B}(A)$-comodules $\Delta_{L}:V\to\pi_{B}(A)\otimes V$,
endowed with a right $B$-module structure satisfying the identity
$\Delta_{L}(vb)=v_{(-1)}\pi_{B}(b_{(1)})\otimes v_{(0)}b_{(2)}$, for all $v\in
V,\,b\in B$, and whose morphisms are left $\pi_{B}(A)$-comodule maps and right
$B$-module maps.

Consider the functor
$\Phi:{}^{A}_{B}\mathrm{Mod}_{B}\to{}^{\pi_{B}\,}\mathrm{Mod}_{B}$, given by
$\Phi(\mathcal{F}):=\mathcal{F}/B^{+}\mathcal{F}$, where the left
$\pi_{B}(A)$-comodule structure of $\Phi(\mathcal{{\mathcal{F}}})$ is given by
$\Delta_{L}[f]:=\pi_{B}(f_{(-1)})\otimes[f_{(0)}],$ with square brackets
denoting the coset of an element in $\Phi(\mathcal{{\mathcal{F}}})$. In the
other direction, we define a functor
$\Psi:{}^{\pi_{B}\,}\mathrm{Mod}_{B}\to{}^{A}_{B}\mathrm{Mod}_{B}$ using the
cotensor product $\square_{\pi_{B}(A)}$, which we find convenient to denote by
$\square_{\pi_{B}}$. Setting $\Psi(V):=A\,\square_{\pi_{B}}V$, where the left
$A$-comodule structure of $\Psi(V)$ is defined on the first tensor factor, the
right $B$-module structure is the diagonal one, and if $\gamma$ is a morphism
in ${}^{\pi_{B}\,}\mathrm{Mod}_{B}$, then
$\Psi(\gamma):=\mathrm{id}\otimes\gamma$.

An adjoint equivalence of categories between ${}^{A}_{B}\mathrm{Mod}_{B}$ and
${}^{\pi_{B}\,}\mathrm{Mod}_{B}$ is given by the functors $\Phi$ and $\Psi$,
and the unit natural isomorphism

$\displaystyle\mathrm{U}:{\mathcal{F}}\to\Psi\circ\Phi({\mathcal{F}}),$
$\displaystyle f\mapsto f_{(-1)}\otimes[f_{(0)}].$

We call this _Takeuchi’s equivalence_. The _dimension_
$\mathrm{dim}({\mathcal{F}})$ of an object
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}_{B}$ is the vector space dimension of
$\Phi({\mathcal{F}})$. As established in [44, §1], for any
$\mathcal{F}\in{}^{A}_{B}\mathrm{Mod}$, an isomorphism in the category
${}^{A}_{A}\mathrm{Mod}$ is given by

(14) $\displaystyle\alpha_{\mathcal{F}}:A\otimes_{B}\mathcal{F}\to
A\otimes\Phi(\mathcal{F}),$ $\displaystyle a\otimes f\mapsto
af_{(-1)}\otimes[f_{(0)}].$

Consider next the full subcategory ${}^{A}_{B}\textrm{Mod}_{0}$ of
${}^{A}_{B}\mathrm{Mod}_{B}$ consisting of those objects ${\mathcal{F}}$ which
satisfy $B^{+}{\mathcal{F}}={\mathcal{F}}B^{+}$. The corresponding full
subcategory ${}^{\pi_{B}\,}\mathrm{Mod}_{0}$ of
${}^{\pi_{B}\,}\mathrm{Mod}_{B}$ consists of those objects with the trivial
right $B$-action, and is of course equivalent to the category of left
$\pi_{B}(A)$-comodules. Both sub-categories come equipped with evident
monoidal structures. Indeed, Takeuchi’s equivalence restricts to a monoidal
equivalence between the subcategories, see [30, §4] for further details.

###  4.2. Some Remarks on Quantum Homogeneous Tangent Spaces

In this subsection we recall the generalisation of the results of §2.2 from
Hopf algebras to quantum homogeneous spaces.

Let $A$ be a Hopf algebra, and $W\subseteq A^{\circ}$ and Hopf subalgebra of
$A^{\circ}$, such that

$B:={}^{W}A=\\{b\in B\,|\,b_{(1)}\langle b_{(2)},w\rangle\\}$

a quantum homogeneous $A$-space, and denote by $B^{\circ}$ its dual coalgebra.
A _tangent space_ for $B$ is a subspace $T\subseteq B^{\circ}$ such that
$T\oplus\mathbb{C}1$ is a right coideal of $B^{\circ}$ and $WT\subseteq T$.
For any tangent space $T$, a right $B$-ideal of
$B^{+}:=B\cap\mathrm{ker}(\varepsilon)$ is given by

$\displaystyle I:=\big{\\{}x\in B^{+}\,|\,X(x)=0,\textrm{ for all }X\in
T\big{\\}},$

meaning that the quotient $V^{1}(B):=B^{+}/I$ is naturally an object in the
category ${}^{\pi_{B}}\mathrm{Mod}_{B}$. We call $V^{1}(B)$ the _cotangent
space_ of the fodc. Consider now the object

$\displaystyle\Omega^{1}(B):=A\square_{\pi_{B}}V^{1}(B).$

If $\\{X_{i}\\}_{i=1}^{n}$ is a basis for $T$, and $\\{e_{i}\\}_{i=1}^{n}$ is
the dual basis of $V^{1}(B)$, then the map

$\displaystyle\mathrm{d}:A\to\Omega^{1}(B),$ $\displaystyle
a\mapsto\sum_{i=1}^{n}(X_{i}\triangleright a)\otimes e_{i}$

is a derivation, and the pair $(\Omega^{1}(B),\mathrm{d})$ is a left
$A$-covariant fodc over $B$. This gives a bijective correspondence between
isomorphism classes of finite-dimensional tangent spaces and finitely-
generated left $A$-covariant fodc [17].

Let $T_{A}$ be a finite-dimensional tangent space for the Hopf algebra $A$,
with associated fodc $\Omega^{1}(A)$. The restriction of $\Omega^{1}(A)$ to
$B$ is clearly a left $A$-covariant fodc, which we denote by $\Omega^{1}(B)$.
Moreover, as shown in [17, Corollary 9], the corresponding tangent space is
given by $WT_{A}|_{B}\subseteq B^{\circ}$. We note that the obvious map from
$V^{1}(B)$, the cotangent space of $\Omega^{1}(B)$, to $\Lambda^{1}(A)$, the
cotangent space of $\Omega^{1}(A)$, is injective if and only if
$WT_{A}|_{B}=T_{A}|_{B}$. Finally, let $\\{X_{i}\\}_{i=1}^{n}$ be a basis for
$T_{A}$, and denote by $\\{e_{i}\\}_{i=1}^{n}$ the corresponding dual basis of
$\Lambda^{1}(A)$. Assume that a basis of the tangent space of the restricted
fodc is given by the elements $\\{X_{i}\\}_{i\in J}$, for some subset
$J\subseteq\\{1,\dots,n\\}$, where by abuse of notation, we do not distinguish
notationally between an element $X_{i}\in T_{A}$ and its restriction to an
element of $B^{\circ}$. It now follows that a basis for the image of
$V^{1}(B)$ in $\Lambda^{1}(A)$ is given by $\\{e_{i}\\}_{i\in J}$.

###  4.3. Lusztig Differential Calculi for the $A$-Series Quantum Flag
Manifolds

For $S$ a proper subset of the set of simple roots $\Pi$, consider the Hopf
subalgebra

$\displaystyle
U_{q}(\mathfrak{l}_{S}):=\big{<}K_{i},E_{j},F_{j}\,|\,i=1,\ldots,n;\,j\in
S\big{>}\subseteq U_{q}(\mathfrak{sl}_{n+1}).$

We denote $S^{c}:=\Pi\backslash S$, and represent $S^{c}$ graphically by
coloured nodes in the Dynkin diagram of $\mathfrak{sl}_{n+1}$. In our
description of quantum tangent spaces below, we find it convenient to consider
the sets of weights

$\displaystyle\Delta_{S}^{+}:=(\mathbb{Z}_{\geq 0}\,S)\cap\Delta^{+},$
$\displaystyle\overline{\Delta^{+}_{S}}:=\Delta^{+}\setminus\Delta_{S}^{+},$

where as usual $\Delta^{+}$ denotes the set of positive roots of
$\mathfrak{sl}_{n+1}$. Consider also the coideal subalgebra of
$U_{q}(\mathfrak{l}_{S})$-invariants

$\displaystyle\mathcal{O}_{q}\big{(}\mathrm{SU}_{n+1}/L_{S}\big{)}:={}^{U_{q}(\mathfrak{l}_{S})}\mathcal{O}_{q}(\mathrm{SU}_{n+1}),$

with respect to the natural left $U_{q}(\mathfrak{sl}_{n+1})$-module structure
on $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. As is well-known,
$\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$ is a quantum homogeneous space (see
[6, §5.4] for a detailed explanation of this fact in the notation of this
paper). We call $\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$ the quantum flag
manifold associated to $S$.

When $S^{c}$ consists of a single simple root $\alpha_{r}$, we call the
corresponding quantum flag manifold the _quantum $r$-plane Grassmannian_, and
denote it by $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$. It follows immediately
from the definition of the algebra $\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$
that it contains as subalgebras those quantum Grassmannians
$\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ such that $\alpha_{r}\in S^{c}$.
Moreover, it can be shown these subalgebras generate
$\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$ as an algebra.

Consider now an arbitrary algebra $A$ endowed with a dc $\Omega^{\bullet}(A)$.
For any subalgebra $B\subseteq A$, the sub-$B$-bimodule of
$\Omega^{\bullet}(A)$ generated by elements of the form $\mathrm{d}b$, for
$b\in B$, forms a dc over $B$. We call this dc the _restriction_ of
$\Omega^{\bullet}(A)$ to $B$. Clearly, we have an analogous notion of
restriction for fodc.

######  Definition 4.1.

For a choice of simple roots $S\subseteq\Pi$, we denote by

$\Big{(}\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1}/L_{S}),\leavevmode\nobreak\
\overline{\partial}\Big{)}$

the restriction to $\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$ of the dc
$\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})$, and call it the _Lusztig dc_ of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1}/L_{S})$. Moreover, we denote by
$T^{(0,1)}_{S}$, and $V^{(0,1)}_{S}$, the associated tangent space, and
cotangent space, respectively.

In the following subsection we will investigate these dc for the special case
of the quantum Grassmannians and show that they coincide with the celebrated
Heckenberger–Kolb dc. Following this, we examine in greater detail the Lusztig
dc of the full quantum manifold and prove a direct $q$-deformation of the
classical Borel–Weil theorem. The remaining quantum flag manifolds will be
treated in a subsequent work.

###  4.4. The Heckenberger–Kolb FODC for the Quantum Grassmannians

We say that a dc over a quantum homogeneous $A$-space $B$ is _irreducible_ if
it is simple as an $(A,B)$-relative Hopf module. In [18] Heckenberger and Kolb
showed that, for any simple root $\alpha_{r}\in\Delta^{+}$, the quantum
Grassmannian $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ admits precisely two non-
isomorphic left $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-covariant finite-
dimensional irreducible fodc. We call them the _holomorphic_ , and _anti-
holomorphic_ , _Heckenberger–Kolb fodc_ , and denote them respectively by

$\displaystyle\Omega^{(1,0)}_{q}(\mathrm{Gr}_{n+1,r}),$
$\displaystyle\Omega^{(0,1)}_{q}(\mathrm{Gr}_{n+1,r}).$

Their respective tangent spaces are given by

$\displaystyle T_{\alpha_{r}^{c}}^{(1,0)}:=U_{q}(\mathfrak{l}_{S})F_{r},$
$\displaystyle T_{\alpha_{r}^{c}}^{(0,1)}:=U_{q}(\mathfrak{l}_{S})E_{r}.$

For $S^{c}=\\{\alpha_{r}\\}$, it holds that

$\displaystyle\overline{\Delta^{+}_{S}}=\Big{\\{}\alpha_{i}+\cdots+\alpha_{r}+\cdots+\alpha_{j}\,|\,1\leq
j\leq r\leq n\Big{\\}}.$

As shown in [19, §3], we have the identity

(15) $\displaystyle
T_{\alpha_{r}^{c}}^{(0,1)}=\mathrm{span}_{\mathbb{C}}\Big{\\{}E_{\gamma}\leavevmode\nobreak\
|\leavevmode\nobreak\ \gamma\in\overline{\Delta^{+}_{S}}\Big{\\}},$

with an analogous identity for $T_{\alpha_{r}^{c}}^{(1,0)}$. Moreover, as
shown in [19, §3.3], the maximal prolongations of the two Heckenberger–Kolb
fodc have classical dimension, that is

$\displaystyle\mathrm{dim}\\!\left(\Omega^{(0,k)}_{q}(\mathrm{Gr}_{n+1,r})\right)=\mathrm{dim}\\!\left(\Omega^{(k,0)}_{q}(\mathrm{Gr}_{n+1,r})\right)=\binom{(n+1-r)r}{k}$

for all $k=0,\dots,(n+1-r)r$. We now observe that our fodc over
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ restricts to the anti-holomorphic
Heckenberger–Kolb dc.

######  Proposition 4.2.

The restriction of $\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1)})$ to
$\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ is the anti-holomorphic
Heckenberger–Kolb fodc, for all $\alpha_{r}\in\Delta^{+}1$. Moreover, the
image of $V^{(0,1)}_{\alpha_{r}^{c}}$ in $\Lambda^{(0,1)}_{q}$ is spanned by
the basis elements $e_{\gamma}$, for $\gamma\in\overline{\Delta^{+}_{S}}$.

###### Proof.

For any $\gamma\in\Delta^{+}_{S}$, the root vector $E_{\gamma}$ is a linear
combination of elements of $U_{q}(\mathfrak{l}_{S})$. Thus we have that
$\langle E_{\gamma},b\rangle=0$, for any
$b\in\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})^{+}$, meaning that the image of
$E_{\gamma}$ in $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})^{\circ}$ is zero. We now
see that the restriction of $T^{(0,1)}$ (which we recall was defined in
Corollary 3.4) to the quantum Grassmannian, is spanned by the cosets, in
$\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})\circ$, of the tangent vectors
$E_{\gamma}$, for $\gamma\in\overline{\Delta^{+}_{S}}$. From the remarks at
the end of §4.2, we see that the tangent space of the restriction fodc is
given by

$U_{q}(\mathfrak{l}_{S})\Big{\\{}E_{\gamma}\,|\,\gamma\in\overline{\Delta^{+}_{S}}\Big{\\}}=U_{q}(\mathfrak{l}_{S})T^{(0,1)}_{\alpha_{r}^{c}}=T^{(0,1)}_{\alpha_{r}^{c}},$

where in the penultimate identity we have used the description of the
Heckenberger–Kolb tangent space given in (15), and in the last identity we
have used the fact since $T^{(0,1)}_{\alpha_{r}^{c}}$ is a tangent space, it
must be closed under the action of $U_{q}(\mathfrak{l}_{S})$. Finally, it now
follows from the discussion at the end of §4.2, that the elements
$e_{\gamma}$, for $\gamma\in\overline{\Delta^{+}_{S}}$, for a basis of
$V^{(0,1)}_{\alpha_{r}^{c}}$ in $\Lambda^{(0,1)}$. ∎

######  Example 4.3.

For the case of $A_{3}$, we have three simple roots
$\\{\alpha_{1},\,\alpha_{2},\,\alpha_{3}\\}$ to choose from. For the choice
$S=\\{\alpha_{2},\alpha_{3}\\}$, we get the quantum projective $3$-space
$\mathcal{O}_{q}(\mathbb{C}\mathrm{P}^{3})$, whose tangent space is given
explicitly by

$T_{\alpha_{1}^{c}}^{(0,1)}=\Big{\\{}E_{3}E_{1}E_{2},\,E_{3}E_{2},\,E_{3}\Big{\\}}\subseteq\mathcal{O}_{q}(\mathbb{C}\mathrm{P}^{3})^{\circ},$

where we have used the fact that the tangent space is contained in the dual of
$\mathcal{O}_{q}(\mathbb{C}\mathrm{P}^{3})$, to simplify the Lusztig root
vectors. We can represent this pictorially as

where the first diagram has coloured nodes when the corresponding basis
element of $\Lambda^{(0,1)}_{q}(\mathrm{SU}_{4})$ is contained in the
cotangent space of $\mathcal{O}_{q}(\mathbb{C}\mathrm{P}^{3})$, and the second
diagram is the coloured Dynkin diagram corresponding to the choice of simple
crossed root $S=\\{\alpha_{1}\\}$. The choice $S=\\{\alpha_{1},\alpha_{2}\\}$
gives another copy of $\mathcal{O}_{q}(\mathbb{C}\mathrm{P}^{3})$, and it
admits a similar description. Finally, the choice
$S=\\{\alpha_{1},\alpha_{3}\\}$ gives the first quantum Grassmannian which is
not a quantum projective space, that is to say,
$\mathcal{O}_{q}(\mathrm{Gr}_{4,2})$. Its tangent space is given explicitly by

$\displaystyle
T_{\alpha_{2}^{c}}^{(0,1)}:=\Bigg{\\{}\begin{array}[]{ll}E_{1}E_{2},&E_{2},\\\
E_{3}E_{1}E_{2},&E_{3}E_{2}\end{array}\Big{\\}}\subseteq\mathcal{O}_{q}(\mathrm{Gr}_{4,2})^{\circ}.$

We can represent this pictorially as

where again the first diagram has coloured nodes when the corresponding basis
element is contained in the cotangent space of
$\mathcal{O}_{q}(\mathrm{Gr}_{4,2})$, and the second diagram is the crossed
Dynkin diagram corresponding to the choice of simple crossed root.

######  Example 4.4.

Let $\alpha_{r}\in\Pi$, and let $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ be the
associated quantum Grassmannian. We note that the subspace
$T:=\mathbb{C}E_{r}$ is a tangent space for
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. The restriction of the associated fodc
to $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ has as its tangent space

$T^{(0,1)}_{\alpha_{r}^{c}}=U_{q}(\mathfrak{l}_{S})E_{r}.$

In other words, the fodc on $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ restricts to
the Heckenberger–Kolb anti-holomorphic fodc. In particular, we see that for
$n>2$,

$\mathrm{dim}(T^{(0,1)}_{\alpha_{r}^{c}})>\mathrm{dim}(T),$

meaning that we cannot have an embedding of cotangent spaces.

###  4.5. Some General Lemmas

In this subsection we make some general observations so as to help elucidate
the delicate issue of embedding higher forms on a quantum homogeneous space
$B\subseteq A$ into higher forms on the Hopf algebra $A$ itself. We begin with
a simple technical observation that we find instructive to highlight as a
lemma.

######  Lemma 4.5.

Take an object $\mathcal{F}\in{}^{A}_{B}\mathrm{Mod}$, and an object
$\mathcal{M}\in{}^{A}_{A}\mathrm{Mod}$, which we consider as an object in
${}^{A}_{B}\mathrm{Mod}$ with respect to the obvious forgetful functor from
${}^{A}_{B}\mathrm{Mod}$ to ${}^{A}_{B}\mathrm{Mod}$. For a morphism
$\xi:\mathcal{F}\to\mathcal{M}$ in the category ${}^{A}_{B}\mathrm{Mod}$, we
consider the linear map

$\displaystyle\widehat{\xi}:\Phi(\mathcal{F})\to F(\mathcal{M}),$
$\displaystyle[f]\mapsto[\xi(f)].$

If $\widehat{\xi}$ is an embedding then $\xi$ is also an embedding.

###### Proof.

Since $\xi$ is by assumption a left $B$-module map we have that

$\xi(B^{+}\mathcal{F})\subseteq B^{+}\xi(\mathcal{F})\subseteq
A^{+}\mathrm{M}.$

Thus we see that $\widehat{\xi}$ is well defined. Commutativity of the diagram
of $B$-module maps

follows directly from the formulae for $\mathrm{U}_{A}$ and $\mathrm{U}_{B}$.
Thus we see that if $\widehat{\xi}$ is injective, then $\xi$ is also
injective. ∎

We note that the converse to this result is not true. Explicitly, if $\xi$ is
an embedding then it is not necessarily true that $\widehat{\xi}$ is an
embedding. An example is given by Example 4.4.

######  Lemma 4.6.

Let $B\subseteq A$ be a quantum homogeneous space, $\Omega^{\bullet}(A)$ a dc
over $A$, and $\Omega^{\bullet}(B)$ a dc over $B$. Let
$p:\Omega^{\bullet}(B)\to\Omega^{\bullet}(A)$ be a morphism in
${}^{A}_{B}\mathrm{Mod}$, and assume that $Ap(\Omega^{1}(B))$ is a right
$A$-submodule of $\Omega^{1}(A)$. Then the image of the function

$\displaystyle\widehat{p}:\Phi(\Omega^{\bullet}(B))\to
F(\Omega^{\bullet}(A)),$

is a subalgebra of $F(\Omega^{\bullet}(A))$.

###### Proof.

We have assumed that $Ap(\Omega^{1}(B))$ is a right $A$-submodule of
$\Omega^{1}(A)$. Let us also assume that $Ap(\Omega^{k}(B))$ is a right
$A$-submodule of $\Omega^{1}(A)$. Then

$\displaystyle Ap(\Omega^{k+1}(B))A=$ $\displaystyle Ap(\Omega^{k})\wedge
p(\Omega^{1}(A))A$ $\displaystyle\subseteq$ $\displaystyle
Ap(\Omega^{k})A\wedge p(\Omega^{1}(A))$ $\displaystyle\subseteq$
$\displaystyle Ap(\Omega^{k})\wedge p(\Omega^{1}(A))$ $\displaystyle=$
$\displaystyle Ap(\Omega^{k+1}).$

Thus it follows from an inductive argument that $Ap(\Omega^{\bullet}(B))$ is a
right $A$-submodule of the dc $\Omega^{\bullet}(A)$.

Let us next note that, for any $\omega\in\Omega^{\bullet}(B)$, and any $a\in
A$, it holds that

$[ap(\omega)]=\varepsilon(a)[p(\omega)]\in\mathrm{im}(\widehat{p}).$

Thus, we see that $F(Ap(\Omega^{\bullet}(B)))$ is equal to the image of
$\widehat{p}$.

Since $\Omega^{\bullet}(A)$ is an algebra object in the the category
${}^{A}_{A}\mathrm{Mod}_{A}$, its image under $F$ is an algebra object in
$\mathrm{Mod}_{A}$. Explicitly, the multiplication $m$ is given by

(16) $\displaystyle m\big{(}[\omega]\otimes[\nu]\big{)}=[\omega
S(\nu_{(-1)})\wedge\nu_{(0)}],$

where we have used the monoidal structure map given in (3). We claim that the
image of $\widehat{p}$ is a subalgebra of
$F\big{(}\Omega^{\bullet}(A)\big{)}$. To show this consider the case where
$\omega,\nu\in\Omega^{\bullet}(B)$. Since $Ap(\Omega^{\bullet}(B))$ is a right
$A$-submodule of $\Omega^{\bullet}(A)$, we have that

$p(\omega)S(\nu_{(-1)})\wedge p(\nu_{(0)})\in A\Omega^{\bullet}(B).$

Hence the corresponding coset in $F(\Omega^{\bullet}(A))$ is contained in the
subspace $F(A\Omega^{\bullet}(B))$, which is to say, it is contained in the
image of $\widehat{p}$. Thus we have a subalgebra as claimed. ∎

###  4.6. Embedding the Higher-Order Forms

As we saw in §4.4, the restriction of our fodc over
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ to the quntum Grassmannians coincides
with the Heckenberger–Kolb fodc. We will now extend this result to higher
forms. However, it is important to note that in general, an inclusion of fodc
does not imply an inclusion of higher-order forms. This is illustrated in
Example 4.9 below.

######  Lemma 4.7.

The subspace
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})\Omega^{(0,1)}_{q}(\mathrm{Gr}_{n+1,r})$ is
a right $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-submodule of
$\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})$.

###### Proof.

It follows from the description of the right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module structure of $\Lambda^{(0,1)}_{q}$
given in (6), and the description of the image of $V^{(0,1)}_{\alpha_{r}^{c}}$
in $\Lambda^{(0,1)}$ given in Proposition 4.2, that the image is a right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-submodule. Now from the definition of the
action of $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ on

$\displaystyle\mathcal{O}_{q}(\mathrm{SU}_{n+1})\Omega^{(0,1)}_{q}(\mathrm{Gr}_{n+1,r})\simeq\mathcal{O}_{q}(\mathrm{SU}_{n+1})\otimes
V^{(0,1)}_{\alpha_{r}^{c}},$

we see that we do indeed have a right sub-module. ∎

######  Proposition 4.8.

The unique dga map $p$ from $\Omega^{(0,\bullet)}_{q}(\mathrm{Gr}_{n+1,r})$ to
$\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})$ induces an isomorphism between
the Lusztig dc and the Heckenberger–Kolb dc of the quantum Grassmannians.

###### Proof.

It follows from Lemma 4.6, and Lemma 4.7 above, that the image of
$\widehat{p}$ is a subalgebra of
$F\big{(}\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})\big{)}$. It was shown in
Proposition 4.2 that, for all $\gamma\in\overline{\Delta^{+}_{S}}$, the basis
element $e_{\gamma}$ is in the image of $\widehat{p}$. Hence the image of
$\widehat{p}$ contains all basis elements of the form

$\displaystyle\Big{\\{}e_{\gamma_{1}}\wedge\cdots\wedge
e_{\gamma_{k}}\,|\,\gamma_{1}<\cdots<\gamma_{k}\in\overline{\Delta^{+}_{S}}\Big{\\}}.$

This means that the image of $\widehat{p}$ has dimension at least as large as
the quantum exterior algebra of the Heckenberger–Kolb dc, meaning that
$\widehat{p}$ must be an embedding. Lemma 4.5 now implies that the dga map $p$
from the Heckenberger–Kolb dc to the Lusztig dc is an isomorphism. ∎

We finish with an example that demonstrates why one needs to be careful when
considering maximal prolongations, even in the case of covariant fodc over
quantum homogeneous spaces.

######  Example 4.9.

For the special case of $\mathcal{O}_{q}(\mathrm{SU}_{2})$, we have a single
non-empty subset $S$ of $\Pi$, namely $\Pi$ itself. The associated quantum
flag manifold is the celebrated _quantum projective line_
$\mathcal{O}_{q}(\mathbb{CP}^{1})$, also known as the _Podleś sphere_ and
denoted by $\mathcal{O}_{q}(S^{2})$. The Podleś sphere is the simplest example
of a quantum Grassmannian, and its Heckenberger–Kolb dc is the _Podleś dc_
orininally discovered in [34], and classified it with respect to a natural set
of requirements. Indeed, Heckenberger–Kolb’s construction and classification
theorems can be considered as a vast generallisation of the work of Podleś.

Consider now the following subspace of $U_{q}(\mathfrak{sl}_{2})$:

$\displaystyle T:=\mathrm{span}_{\mathbb{C}}\\{F,\,E\\}.$

It follows directly from the coproduct formulae of $U_{q}(\mathfrak{sl}_{2})$
that $T$ is a tangent space for $\mathcal{O}_{q}(\mathrm{SU}_{2})$. Moreover,
we see that it is closed under the left action of $U_{q}(\mathfrak{h})$,
meaning that the tangent space of the restriction to the Podleś sphere
$\mathcal{O}_{q}(S^{2})$ is spanned by the cosets of $E$ and $F$ in
$\mathcal{O}_{q}(S^{2})^{\circ}$. This associated fodc is of course the Podleś
fodc. Denoting by $\\{f,e\\}$ the dual basis of the cotangent space, it holds
that

$\Omega^{2}_{q}(S^{2})=\mathcal{O}_{q}(S^{2})f\wedge e,$

and that all higher forms are zero, as explained for example in [28, §1].

Let us now consider the maximal prolongation of
$\Omega^{1}_{q}(\mathrm{SU}_{2})$, the fodc associated to $T$. We note that
the corresponding ideal $I\subseteq\mathcal{O}_{q}(\mathrm{SU}_{2})^{+}$
contains the elements $u^{+}_{11}$ and $u_{22}^{+}$. Operating by the map
$\omega$ gives

$\displaystyle\omega(u^{+}_{11})=[u_{12}]\otimes[u_{21}]=e\otimes f,$
$\displaystyle\omega(u^{+}_{22})=[u_{21}]\otimes[u_{12}]=e\otimes f,$

meaning that the image of $\Omega^{2}_{q}(S^{2})$ in
$\Omega^{2}_{q}(\mathrm{SU}_{2})$ is zero. Thus we see that even when the fodc
on $A$ is right $\pi_{B}(A)$-covariant, we are not guaranteed an embedding of
the maximal prolongation of $\Omega^{1}(B)$.

##  5\. The Lusztig DC for the Full Quantum Flag Manifolds

In this section we look at the Lusztig dc for the full quantum flag manifolds,
and we see that it is a direct $q$-deformation of the anti-holomorphic
subcomplex of the classical $A$-series full quantum flag manifold. We present
the dc as a direct sum of line bundles over
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$. Notably, we show that its right
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$-module structure is not the trivial one,
as is the case for the Heckenberger–Kolb calculi. Thus it lives outside the
subcategory of those relative Hopf modules for which the simple monoidal
version of Takeuchi’s equivalence applies.

###  5.1. Some Details on Full Quantum Flag Manifolds

We begin by recalling some well-known results about the $A$-series full
quantum flag manifolds. This is mainly to fix notation, and we refer the
reader to [6, §5] for a detailed exposition for this material in the notation
of this paper.

The quantum flag manifold $\mathcal{O}_{q}(\mathrm{F}_{n+1})$ associated to
the choice of simple nodes $S=\varnothing$ is called the _full quantum flag
manifold_ of $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. In this case the isotropy
Hopf algebra is isomorphic to $\mathcal{O}(\mathbb{T}^{n})$, which is to say
the group Hopf algebra of the weight lattice
$\mathcal{P}\simeq\mathbb{Z}^{n}$. Thus the simple objects in
${}^{A}_{B}\mathrm{Mod}_{0}$ are labelled by elements $\lambda$ of the weight
lattice $\mathcal{P}$. As explained in [6, Example 5.5], this means that we
have a $\mathcal{P}$-algebra grading

$\mathcal{O}_{q}(\mathrm{SU}_{n+1})\simeq\bigoplus_{\lambda\in\mathcal{P}}{\mathcal{E}}_{\lambda}.$

Moreover, the grading is _strong_ , which is to say,

$\displaystyle{\mathcal{E}}_{\lambda}\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\mu}\simeq{\mathcal{E}}_{\lambda}{\mathcal{E}}_{\mu}={\mathcal{E}}_{\lambda+\mu},$
$\displaystyle\textrm{ for all }\lambda,\,\mu\in\mathcal{P},$

where the first isomorphism is given by multiplication. Thus we see that
${\mathcal{E}}_{\lambda}$ is an invertible bimodule with inverse
${\mathcal{E}}^{-1}_{\lambda}={\mathcal{E}}_{-\lambda}$. Motivated by
classical geometry, we follow the terminology of [1], we call an invertible
bimodule in ${}^{A}_{B}\mathrm{Mod}_{0}$ a _relative line module_.

As shown in [41, §Theorem 4.1], a set of algebra generators for
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is given by

(17) $\displaystyle z^{\varpi_{x}}_{i}:=c^{\varpi_{x}}_{f_{i},v_{N_{x}}},$
$\displaystyle\overline{z}_{i}^{\,\varpi_{x}}:=c^{-w_{0}(\varpi_{x})}_{v_{i},f_{N_{x}}}$
$\displaystyle\textrm{ for }i=1,\dots,N_{x},\textrm{ and }x=1,\dots,n,$

where $N_{x}:=\dim(V_{\varpi_{x}})$, and $\\{v_{i}\\}_{i=1}^{N_{x}}$, and
$\\{f_{i}\\}_{i=1}^{N_{x}}$, are a choice of dual weight bases of
$V_{\varpi_{x}}$, and $V_{-w_{0}(\varpi_{x})}$ respectively, that we fix for
once and for all. We also find it convenient to introduce the notation

$\displaystyle
z^{\lambda}:=(z^{\varpi_{1}}_{N_{x}})^{a_{1}}\cdots\,(z^{\varpi_{n-1}}_{N_{x}})^{a_{x}},$
$\displaystyle\text{ for any }a_{1},\dots,a_{n}\in\mathbb{Z}_{\geq 0},\textrm{
with }\lambda=:\sum_{x=1}^{n}a_{x}\varpi_{x}\in\mathcal{P}^{+}.$

Finally, we observe that the $\mathcal{P}$-grading of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is determined by
$|z^{\varpi_{x}}_{i}|=\varpi_{x}$, and
$|\overline{z}^{\,\varpi_{x}}_{i}|=-\varpi_{x}$, for all $i=1,\dots,N_{x}$.
Explicitly, for $\lambda\in\mathcal{P}^{+}$, the line bundle
${\mathcal{E}}_{\lambda}$ is generated as a right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module by the subspace
$U_{q}(\mathfrak{sl}_{n+1})z^{\lambda}\subseteq{\mathcal{E}}_{\lambda}.$

###  5.2. A Higher Order Extension of Majid’s Framing Theorem

As we saw from Example 4.4, the contangent space of a covariant first-order
differential calculus over a quantum homogeneous space $B\subseteq A$ does not
necessarily embed into the cotangent space of the Hopf algebra $A$. Necessary
and sufficient conditions for this to happen were discussed at the end of
§4.2. It was proved by Majid in [28, Theorem 2.1] that right
$\pi_{B}(A)$-covariance of the fodc on $\Omega^{1}(A)$ guarantees an
embedding. While this requirement is not necessary, it is quite natural from a
categorical point of view, and it is this version that we now extend to higher
forms.

Consider the category ${}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$ whose objects are
left $A$-Hopf modules $\mathcal{M}$ endowed with a right $\pi_{B}(A)$-comodule
structure $\Delta_{R}:\mathcal{M}\to\mathcal{M}\otimes\pi_{B}(A)$ giving
$\mathcal{M}$ the structure of an $A$-$\pi_{B}(A)$-bicomodule, and for which
$\Delta_{R}(am)=\Delta_{R,\pi_{B}}(a)\Delta_{R}(m)$, for all $a\in
A,m\in\mathcal{M}$.

As observed in [6], we can extend the fundamental theorem of Hopf modules to
an equivalence between ${}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$ and
$\mathrm{Mod}^{\pi_{B}}$, the category of right $\pi_{B}(A)$-comodules: For
$\mathcal{M}\in{}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$, we endow $F(\mathcal{M})$
with the right $\pi_{B}(A)$-coaction
$[m]\mapsto[m_{(0)}]\otimes\pi_{B}(m_{(1)})$. In the other direction, for some
$(V,\Delta_{R})\in\mathrm{Mod}^{\pi_{B}}$, we endow $A\otimes V$ with the
right $\pi_{B}(A)$-comodule tensor product
$\Delta_{R,\pi_{B}}\otimes\Delta_{R}$.

We also have an equivalence between ${}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$ and the
category of relative Hopf modules ${}^{A}_{B}\mathrm{Mod}$. This was observed
in [6, Proposition 3.4] (in the more general setting of principal pairs of
quantum homogeneous spaces). This equivalence builds on Takeuchi’s equivalence
for left $B$-modules [44, Theorem 2], which was later generalised to
Schneider’s equivalence for principal comodule algebras [39, Theorem I]. First
we define a functor

$\displaystyle
A\otimes_{B}-:{}^{A}_{B}\mathrm{Mod}\to{}^{A}_{A}\mathrm{Mod}^{\pi_{B}},$
$\displaystyle{\mathcal{F}}\mapsto A\otimes_{B}{\mathcal{F}},$

where the left $A$-module structure of $A\otimes_{B}{\mathcal{F}}$ is given by
the first tensor factor, the left $A$-comodule structure is the tensor product
coaction $\Delta\otimes\Delta_{L}$, and the right $\pi_{B}(A)$-comodule
structure is given by $\Delta_{R,\pi_{B}}\otimes\mathrm{id}$. The functor acts
on morphisms as $\mathrm{id}\otimes-$. In the other direction, we have the
functor $\mathrm{co}(-)$ which maps $\mathcal{M}$ to its space of coinvariants

$\mathcal{M}^{\mathrm{co}(\pi_{B})}:=\big{\\{}m\in\mathcal{M}\,|\,\Delta_{R}(m)=m\otimes
1\big{\\}},$

and restricts the domain of morphisms.

Finally, just as for any Hopf algebra, the antipode $S$ of $\pi_{B}(A)$ gives
an equivalence

$\displaystyle\mathrm{S}:\mathrm{Mod}^{\pi_{B}}\to{}^{\pi_{B}}\mathrm{Mod},$
$\displaystyle\mathrm{S^{-1}}:{}^{\pi_{B}}\mathrm{Mod}\to\mathrm{Mod}^{\pi_{B}}.$

Combing these equivalences with Takeuchi’s relative Hopf module equivalence
gives us the following diagram in $\mathbf{Cat}$, the category of all
categories:

$\textstyle{{}^{A}_{A}\mathrm{Mod}^{\pi_{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\mathrm{co}(-)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Mod}^{\pi_{B}}}$$\scriptstyle{A\otimes-}$$\scriptstyle{\mathrm{S}^{-1}}$$\textstyle{{}^{A}_{B}\mathrm{Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A\otimes_{B}-}$$\scriptstyle{\Phi}$$\textstyle{{}^{\pi_{B}}\mathrm{Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\mathrm{S}}$

We will now look at some natural isomorphisms associated to this diagram.

######  Proposition 5.1.

A natural isomorphism between the functor $A\otimes_{B}-$ and the functor
$(A\otimes-)\circ\mathrm{S}\circ\Phi$ is given by the components

(18) $\displaystyle\alpha_{{\mathcal{F}}}:A\otimes_{B}{\mathcal{F}}\to
A\otimes\mathrm{S}(\Phi({\mathcal{F}})),$ $\displaystyle a\otimes f\mapsto
af_{(-1)}\otimes[f_{(0)}],$

where $\mathcal{F}$ is an object in ${}^{A}_{B}\mathrm{Mod}$.

###### Proof.

Recall from (14) that (18) gives an isomorphism in the category
${}^{A}_{A}\mathrm{Mod}$. We will now show that this is also a morphism of
right $\pi_{B}(A)$-comodules, which is to say, we will show that the following
diagram commutes:

This follows from the calculation

$\displaystyle(\Delta_{R,\pi_{B}}\otimes\Delta_{R})(\alpha_{{\mathcal{F}}}(a\otimes
f))=$
$\displaystyle\,\big{(}\Delta_{R,\pi_{B}}\otimes\,\Delta_{R})(af_{(-1)}\otimes[f_{(0)}]\big{)}$
$\displaystyle=$
$\displaystyle\,a_{(1)}f_{(-3)}\otimes[f_{(0)}]\otimes\pi_{B}\big{(}a_{(2)}f_{(-2)}S(f_{(-1)})\big{)}$
$\displaystyle=$
$\displaystyle\,a_{(1)}f_{(-1)}\otimes[f_{(0)}]\otimes\pi_{B}(a_{(2)})$
$\displaystyle=$
$\displaystyle\,(\alpha_{{\mathcal{F}}}\otimes\mathrm{id})\circ(\Delta_{R,\pi_{B}}\otimes\mathrm{id})(a\otimes
f).$

It is now readily confirmed that the family of isomorphisms

$\Big{\\{}\alpha_{{\mathcal{F}}}\,|\,{\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}^{\pi_{B}}\Big{\\}}$

give a natural isomorphism. ∎

From this follows a number of other natural isomorphisms. We highlight the
following which will be a key ingredient in the higher order extension of
Majid’s framing theorem below. We omit the proof, which is clear.

######  Corollary 5.2.

A natural isomorphism between the two functors $F\circ(A\otimes_{B}-)$ and
$\mathrm{S}\circ\Phi$ is given by the components

$\displaystyle\beta_{{\mathcal{F}}}:F(A\otimes_{B}{\mathcal{F}})\to\mathrm{S}(\Phi({\mathcal{F}})),$
$\displaystyle[a\otimes f]\mapsto\varepsilon(a)[f],$

where $\mathcal{F}\in{}^{A}_{B}\mathrm{Mod}$.

In the following corollary, we now apply this natural isomorphism to the
question of embedding higher differential forms.

######  Corollary 5.3.

Let $B\subseteq A$ be a quantum homogeneous space, $\Omega^{\bullet}(A)$ a dc
in the category ${}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$, and $\Omega^{\bullet}(B)$
the restriction to a dc on $B$. Then the map

$\displaystyle i:\mathrm{S}\circ\Phi(\Omega^{\bullet}(B))\to
F(\Omega^{\bullet}(A)),$ $\displaystyle[\omega]\mapsto[\omega]$

is a monomorphism in the category $\mathrm{Mod}^{\pi_{B}}$. Moreover, we have
that

$\displaystyle\Omega^{\bullet}(B)\simeq
A\,\square_{\pi_{B}}\mathrm{S}(i(\Phi(\Omega^{\bullet}(B))).$

###### Proof.

Note first that $\Omega^{\bullet}(B)$ is contained in the right coinvariant
forms of $\Omega^{\bullet}(A)$, giving us an embedding

$\Omega^{1}(B)\hookrightarrow(\Omega^{\bullet}(A))^{\mathrm{co}(\pi_{B})}.$

Operating on this inclusion by the functor $A\otimes_{B}-$ gives us the
inclusion

(19) $\displaystyle A\otimes_{B}\Omega^{1}(B)\hookrightarrow
A\otimes_{B}(\Omega^{\bullet}(A))^{\mathrm{co}(\pi_{B})}.$

Considering next the inverse of the unit $\mathrm{U}$ of the equivalence
between ${}^{A}_{A}\mathrm{Mod}^{\pi_{B}}$ and ${}^{A}_{B}\mathrm{Mod}$, we
see that

$\displaystyle\mathrm{U}^{-1}:A\otimes_{B}(\Omega^{\bullet}(A)^{\mathrm{co}(\pi_{B})})\simeq\Omega^{\bullet}(A),$
$\displaystyle a\otimes\omega\mapsto a\omega.$

Combining this with the inclusion in (19), we see that

$\displaystyle
A\otimes_{B}\Omega^{\bullet}(B)\hookrightarrow\Omega^{\bullet}(A),$
$\displaystyle a\otimes\omega\mapsto a\omega.$

Operating on this by the functor $F$, we get the inclusion

$\displaystyle F(A\otimes_{B}\Omega^{\bullet}(B))\hookrightarrow
F(\Omega^{\bullet}(A)).$

Finally, combining this with the appropriate component of the natural
isomorphism given in Corollary 5.2, we get the claimed inclusion ∎

###  5.3. The Lusztig Differential Calculus of
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$

In this subsection we look at the restriction of
$\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})$ to the $A$-series full quantum flag
manifolds, observing that they have classical dimension. Moreover, we also
describe the maximal prolongation of this fodc as a direct sum of relative
line modules.

######  Lemma 5.4.

The embedding
$V^{(0,\bullet)}_{\varnothing}\hookrightarrow\Lambda^{(0,\bullet)}$ is an
isomorphism of $\mathcal{O}_{q}(\mathbb{T}^{n})$-comodules. Hence it holds
that

(20)
$\displaystyle\Omega^{(0,\bullet)}_{q}(\mathrm{F}_{n+1})\leavevmode\nobreak\
=\leavevmode\nobreak\
\mathcal{O}_{q}(\mathrm{SU}_{n+1})\square_{\,\pi_{B}}\Lambda^{(0,\bullet)}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\
\Omega^{(0,\bullet)}_{q}(\mathrm{SU}_{n+1})^{\mathrm{co}(\mathcal{O}(\mathbb{T}^{n})}.$

Moreover, by transference of structure, this isomorphism endows
$V^{(0,\bullet)}_{\varnothing}$ with a right
$\mathcal{O}_{q}(SU_{n+1})$-module structure.

###### Proof.

We recall from Corollary 3.5 that the fodc
$\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})$ is right
$\mathcal{O}_{q}(\mathbb{T}^{n})$-covariant. Thus it follows from Corollary
5.3 that we have an embedding in the category $\mathrm{Mod}^{\pi_{B}}$

(21) $\displaystyle
V^{(0,\bullet)}_{\varnothing}\hookrightarrow\Lambda^{(0,\bullet)},$

which is to say, an embedding of $\mathcal{O}(\mathbb{T}^{n})$-modules. It
follows from the description of the Heckenberger–Kolb tangent space given in
§4.4 that every basis element of $T^{(0,1)}$ pairs non-trivially with some
element of some quantum Grassmannian. Now every quantum Grassmannian
$\mathcal{O}_{q}(\mathrm{Gr}_{n+1,r})$ is contained as a subalgebra of
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$. Thus we see that the embedding of
$V^{(0,1)}_{\varnothing}$ in $\Lambda^{(0,1)},$ is in fact an isomorphism.
Moreover, following the argument of Proposition 4.8, we can show that the
image of $V^{(0,\bullet)}_{\varnothing}$ is a subalgebra of
$\Lambda^{(0,\bullet)}$. Thus we see that (21) is in fact an isomorphism. The
identities in (20) now follows directly, as does the induced
$\mathcal{O}_{q}(SU_{n+1})$-module structure on
$V^{(0,\bullet)}_{\varnothing}$. ∎

For the case of full quantum flag manifolds, we recall that the isotropy Hopf
algebra is $\mathcal{O}(\mathbb{T}^{n})$. This means that any finite-
dimensional comodule of the isotropy Hopf algebra is one-dimensional, and
hence invertible. Equivalently, every relative Hopf module
$\mathcal{F}\in{}^{A}_{B}\mathrm{Mod}_{0}$ decomposes into a direct sum of
line modules. The following proposition describes the decompositions of our
space of $k$-forms into line modules.

######  Proposition 5.5.

We have an isomorphism, in the category ${}^{A}_{B}\mathrm{Mod}$,

(22)
$\displaystyle\Omega^{(0,k)}_{q}(\mathrm{F}_{n+1})\simeq\bigoplus_{\gamma_{1}<\cdots<\gamma_{k}}{\mathcal{E}}_{\gamma_{1}+\cdots+\gamma_{k}},$

where summation is over ordered $k$-tuples of positive roots of
$\mathfrak{sl}_{n+1}$.

###### Proof.

It follows from the fact that $\Lambda^{(0,\bullet)}_{q}$ is an
$\mathcal{O}(\mathbb{T}^{n})$-comodule algebra, that every basis element is a
weight vector. Taken together with Corollary 3.5, we now see that

$\displaystyle|e_{\gamma_{1}}\wedge\cdots\wedge
e_{\gamma_{k}}|=\gamma_{1}+\cdots+\gamma_{k}.$

It now follows that

(23) $\displaystyle\Omega^{(0,k)}_{q}(\mathrm{F}_{n+1})\simeq$
$\displaystyle\,\bigoplus_{\gamma_{1}<\cdots<\gamma_{k}}\mathcal{O}_{q}(\mathrm{SU}_{n+1})\square_{\pi_{B}}\mathbb{C}e_{\gamma_{1}}\wedge\cdots\wedge
e_{\gamma_{k}}$ (24) $\displaystyle\simeq$
$\displaystyle\bigoplus_{\gamma_{1}<\cdots<\gamma_{k}}{\mathcal{E}}_{\gamma_{1}+\cdots+\gamma_{k}},$

where summation is over ordered $k$-tuples of positive roots of
$\mathfrak{sl}_{n+1}$. ∎

As we will see in the next subsection, the Lusztig dc does not live in
subcategory ${}^{A}_{B}\mathrm{Mod}_{0}$. This means that we do not have
access to the convenient monoidal form of Takeuchi’s equivalence, as treated
in [30]. This means that in order to describe forms twisted by line bundles,
we need some new tools.

######  Lemma 5.6.

Let $B\subseteq A$ be a quantum homogeneous space, and
${\mathcal{F}},\mathcal{G}\in{}^{A}_{B}\mathrm{Mod}_{B}$. Then an isomorphism
in the category ${}^{\pi_{B}}\mathrm{Mod}$ is given by

$\displaystyle\psi:\Phi({\mathcal{F}}\otimes_{B}\mathcal{G})\to\Phi({\mathcal{F}})\otimes_{B}\mathcal{G},$
$\displaystyle[f\otimes g]\mapsto[f]\otimes g.$

###### Proof.

For $f\in{\mathcal{F}},\,g\in\mathcal{G}$, and $b\in B$, we note that

$\displaystyle[fb]\otimes g=[f]\otimes bg,$ $\displaystyle[bf]\otimes
g=\varepsilon(b)[f]\otimes g.$

From this we can conclude that $\psi$ is well-defined. An inverse is given by

$\displaystyle\psi^{-1}:\Phi({\mathcal{F}})\otimes_{B}\mathcal{G}\to\Phi({\mathcal{F}}\otimes_{B}\mathcal{G}),$
$\displaystyle[f\otimes g]\mapsto[f]\otimes g,$

which is confirmed to be well-defined in an analogous manner. ∎

######  Lemma 5.7.

Let $A,B,{\mathcal{F}}$, and $\mathcal{G}$ be as in the lemma above, and
moreover, assume that $\Phi({\mathcal{F}})$ has a right $A$-action extending
its right $B$-action. Then an isomorphism in the category
${}^{\pi_{B}}\mathrm{Mod}$ is given by

$\displaystyle\phi:\Phi({\mathcal{F}}\otimes_{B}\mathcal{G})\to\Phi({\mathcal{F}})\otimes\Phi(\mathcal{G}),$
$\displaystyle[f\otimes g]\mapsto[f]g_{(-1)}\otimes[g_{(0)}].$

###### Proof.

Just as in the previous lemma, it is routine to check that $\phi$ is well
defined as a morphism in the category ${}^{\pi_{B}}\mathrm{Mod}$. Consider
next the map

$\displaystyle\phi^{-1}:\Phi({\mathcal{F}})\otimes\Phi(\mathcal{G})\to\Phi({\mathcal{F}})\otimes_{B}\mathcal{G},$
$\displaystyle[f]\otimes[g]\mapsto[f]S(g_{(-1)})\otimes g_{(0)}.$

Using the previous lemma to identify
$\Phi({\mathcal{F}}\otimes_{B}\mathcal{G})$ and
$\Phi({\mathcal{F}})\otimes_{B}\mathcal{G}$, we see that $\phi^{-1}$ does
indeed define an inverse to $\phi$, and hence we have an isomorphism. ∎

Using the isomorphism given above, together with the right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module structure of
$V^{(0,1)}_{\varnothing}$ given in Lemma 5.4, we can now describe the space of
forms twisted by a line bundle as a direct sum of line bundles.

######  Proposition 5.8.

For any $\lambda\in\mathcal{P}$, we have an isomorphism, in the category
${}^{A}_{B}\mathrm{Mod}$,

(25)
$\displaystyle\Omega^{(0,k)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\lambda}\simeq\bigoplus_{\gamma_{1}<\cdots<\gamma_{k}}{\mathcal{E}}_{\gamma_{1}+\cdots+\gamma_{k}+\lambda},$

where summation is over ordered $k$-tuples of positive roots of
$\mathfrak{sl}_{n+1}$.

###### Proof.

It follows from Lemma 5.7 that we have an isomorphism

$\displaystyle\Phi\\!\left(\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\lambda}\right)\to\Phi(\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1}))\otimes\Phi({\mathcal{E}}_{\lambda}).$

The claimed isomorphism in (25) now follows from Takeuchi’s equivalence,
Proposition 5.5 and the additivity of weights with respect to tensor products.
∎

###  5.4. The Full Lusztig DC and the Sub-Category
${}^{A}_{B}\mathrm{Mod}_{0}$

It follows from [19, Proposition 3.4] that all the Heckenberger–Kolb anti-
holomorphic fodc are contained in the subcategory
${}^{A}_{B}\mathrm{Mod}_{0}$. Thus in particular, the Podleś fodc is contained
in this subcategory. However, as the following proposition shows, the Podleś
sphere is the only full quantum flag manifold for which this happens.

######  Proposition 5.9.

For $n\geq 2$, the fodc $\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})$ is not an
object in ${}^{A}_{B}\mathrm{Mod}_{0}$.

###### Proof.

Consider the element

$b:=u_{11}u_{32}u_{33}u_{44}\cdots
u_{n+1,n+1}\in\mathcal{O}_{q}(\mathrm{SU}_{n+1}).$

Its weight is given by the sum of the weights occurring in $V_{\varpi_{1}}$,
which is to say, it is zero. Thus we see that
$b\in\mathcal{O}_{q}(\mathrm{F}_{n+1})$.

Acting on $e_{21}\in V^{(0,1)}_{\varnothing}$ by $b$, we get the non-zero
element

$\displaystyle e_{21}b=e_{21}u_{11}u_{32}u_{33}\cdots u_{n+1,n+1}=\,$
$\displaystyle q^{-1}\nu e_{31},$

where the identity $e_{21}u_{32}=\nu e_{31}$ makes sense because $n\geq 2$ by
assumption. Thus we see that $b$ does not act on $e_{21}$ by
$\varepsilon(b)=0$, meaning that $V^{(0,1)}_{\varnothing}$ is not contained in
the subcategory ${}^{\pi_{B}}\mathrm{Mod}_{0}$. It now follows from the
subcategory equivalence between ${}^{\pi_{B}}\mathrm{Mod}_{0}$ and and
${}^{A}_{B}\mathrm{Mod}_{0}$ that the fodc is not contained in the subcategory
${}^{A}_{B}\mathrm{Mod}_{0}$, for $n\geq 2$. ∎

###  5.5. Some Examples and Observations

In this subsection we treat some special cases of the results of the previous
section. Moreover, we make some observations about the fact that the Lusztig
dc of $\mathcal{O}_{q}(\mathrm{F}_{n+1})$ are not contained in the category
$\mathrm{Mod}_{0}$.

######  Example 5.10.

In addition to being the simplest example of a quantum Grassmannian, the
Podleś sphere $\mathcal{O}_{q}(S^{2})$ is also the simplest full quantum-flag
manifold. As shown in [18], and independently in [28], we have a decomposition

$\Omega^{1}_{q}(S^{2})\simeq\Omega^{(1,0)}\oplus\Omega^{(0,1)},$

generalising the decomposition of the complexified $1$-forms of $S^{2}$ into
holomorphic and anti-holomorphic forms. As a comparison of tangent spaces
shows, $\Omega^{(0,1)}$ coincides with $\Omega^{(0,1)}_{q}(S^{2})$.

Recall that the weight lattice of $\mathfrak{sl}_{2}$ is isomorphic to
$\mathbb{Z}$, meaning that we can label the line bundles as
${\mathcal{E}}_{k}$, for $k\in\mathbb{Z}$. The root lattice corresponds to the
sublattice $2\mathbb{Z}$, and we have that $\alpha_{1}=2\varpi_{1}$. From this
we see that

$\Omega^{(0,1)}_{q}(S^{2})\simeq{\mathcal{E}}_{2}$

This reproduces Majid’s quantum frame bundle description of the anti-
holomorphic one-forms of the Podleś calculus [28, Theorem 3.1].

######  Example 5.11.

It follows from the description of the module structure of
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{n+1})$ that the decomposition in Proposition
5.5 is not a decomposition as right
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$-modules. However, the _column_
decomposition

$\displaystyle\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\simeq\bigoplus_{i=1}^{n}M_{i},$
$\displaystyle\textrm{ where
}M_{i}:=\bigoplus_{j=i}^{n}{\mathcal{E}}_{\alpha_{i}+\,\cdots\,+\alpha_{j}},$

is a decomposition in the category ${}^{A}_{B}\mathrm{Mod}_{B}$. We note that
none of the summands are irreducible. Finally, it is interesting to note that

$\bigoplus_{i=1}^{n}{\mathcal{E}}_{\alpha_{i}\,+\cdots\,+\alpha_{n-1}}$

is a sub-bimodule of the fodc, and that it is an object in the sub-category
${}^{A}_{B}\mathrm{Mod}_{0}$.

######  Example 5.12.

For $\mathcal{O}_{q}(\mathrm{F}_{3})$, the $1$-forms are given explicitly by

$\displaystyle\Omega^{(0,1)}_{q}(\mathrm{F}_{3})\simeq{\mathcal{E}}_{\alpha_{1}}\oplus{\mathcal{E}}_{\alpha_{2}}\oplus{\mathcal{E}}_{\alpha_{1}+\alpha_{2}}=M_{1}\oplus
M_{2}.$

For the two forms we see that

$\displaystyle\Omega^{(0,1)}_{q}(\mathrm{F}_{3})\simeq{\mathcal{E}}_{\alpha_{1}+\alpha_{2}}\oplus{\mathcal{E}}_{2\alpha_{1}+\alpha_{2}}\oplus{\mathcal{E}}_{\alpha_{1}+2\alpha_{2}}.$

Finally, the space of top forms is given by the single line bundle

$\Omega^{(0,3)}_{q}(\mathrm{F}_{3})\simeq{\mathcal{E}}_{2(\alpha_{1}+\alpha_{2})}={\mathcal{E}}_{2\rho},$

where $\rho$ is the half-sum of positive roots of $\mathfrak{sl}_{3}$. It
follows from the formula for the $\mathcal{O}_{q}(\mathrm{F}_{3})$-module
structure of $V^{(0,1)}_{\varnothing}$ that the space of top forms is
contained in the subcategory ${}^{A}_{B}\mathrm{Mod}_{0}$. We note that the
space of top forms is non-trivial, which is to say we do not have isomorphism
${\mathcal{E}}_{0}\simeq\Omega^{(0,3)}(\mathrm{F}_{3})$, or in other words,
the dc is not _orientable_. In the classical case, a Kähler manifold whose
space of top anti-holomorphic forms is a trivial line bundle is called a
_Calabi–Yau manifold_. While every flag manifold is a Kähler manifold, _n_ o
flag manifold is a Calabi–Yau manifold.

######  Example 5.13.

For the general $\mathcal{O}_{q}(\mathrm{F}_{n+1})$ case, the space of top
anti-holomorphic forms is given by

$\Omega^{(0,|\Delta^{+}|)}_{q}(\mathrm{F}_{n+1})\simeq{\mathcal{E}}_{2\rho}\in{}^{A}_{B}\mathrm{Mod}_{0}.$

Again we see that the dc is not orientable, generalising the fact that
$\mathrm{F}_{n+1}$ is not a Calabi–Yau manifold.

##  6\. A Borel–Weil Theorem for the Full Quantum Flag Manifolds

In this section we construct a direct quantum group generalisation of the
classical Borel–Weil theorem for the $A$-series full flag manifold. This
gives, for the first time, a noncommutative differential geometric realisation
of all the type-$1$ finite-dimensional irreducible representations of
$U_{q}(\mathfrak{sl}_{n+1})$. This approach can easily be adapted to treat the
quantum Grassmannians $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,m})$, whereupon it
reproduces the quantum Borel–Weil theorem given in [5] and [4]. We note that
the existence of a tractable fodc on the Hopf algebra
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ allows for a significant simplification
of these earlier proofs.

###  6.1. Connections and Quantum Principal Bundles

For $\Omega^{1}$ a dc over an algebra $B$, and $\mathcal{F}$ a left
$B$-module, a _left affine connection_ for ${\mathcal{F}}$ is a
$\mathbb{C}$-linear map
$\nabla:\mathcal{F}\to\Omega^{1}\otimes_{B}{\mathcal{F}}$ satisfying

(26) $\displaystyle\nabla(bf)=\mathrm{d}b\otimes f+b\nabla f,$
$\displaystyle\textrm{ for all }b\in B,f\in{\mathcal{F}}.$

The difference of two affine connections $\nabla-\nabla^{\prime}$ is a left
$B$-module map. Any affine connection can be extended to a map
$\nabla:\Omega^{\bullet}\otimes_{B}\mathcal{F}\to\Omega^{\bullet}\otimes_{B}\mathcal{F}$
by the formula

$\displaystyle\nabla(\omega\otimes f)=\mathrm{d}\omega\otimes
f+(-1)^{|\omega|}\,\omega\wedge\nabla f,$

where $f\in{\mathcal{F}}$, and $\omega$ is a homogeneous element of
$\Omega^{\bullet}$ of degree $|\omega|$. The _curvature_ of an affine
connection is the left $B$-module map
$\nabla^{2}:\mathcal{F}\to\Omega^{2}\otimes_{B}\mathcal{F}$. An affine
connection is said to be flat if $\nabla^{2}=0$. Since
$\nabla^{2}(\omega\otimes f)=\omega\wedge\nabla^{2}(f)$, an affine connection
is flat if and only if the pair
$(\Omega^{\bullet}\otimes_{B}{\mathcal{F}},\nabla)$ is a cochain complex.

A _left bimodule affine connection_ on ${\mathcal{F}}$ is a pair
$(\nabla,\sigma)$ where $\nabla$ is a left affine connection and
$\sigma:{\mathcal{F}}\otimes_{B}\Omega^{1}(B)\rightarrow\Omega^{1}(B)\otimes_{B}{\mathcal{F}}$
is a bimodule map satisfying

(27) $\displaystyle\sigma(f\otimes db)=\nabla(fb)-\nabla(f)b.$

Note that in the commutative case $\sigma$ reduces to the standard flip map.

We now briefly recall Brzezinski and Majid’s theory of quantum principal
bundles, a framework for constructing affine connections. To keep the
preliminaries to a minimum, we restrict to the quantum homogeneous space case.
For more details, we direct the interested reader to [1, §5] or to [4, §2]. A
_homogeneous quantum principal bundle_ is a pair $(B,\Omega^{1}(A))$
consisting of a quantum homogeneous space $B\subseteq A$, and a left
$A$-covariant, right $\pi_{B}(A)$-covariant fodc $\Omega^{1}(A)$ over $A$. A
_principal connection_ for $(B,\Omega^{1}(A))$ is a left $A$-module, right
$\pi_{B}(A)$-comodule, map

$\Pi:\Omega^{1}(A)\to A\Omega^{1}(B)A,$

satisfying $\Pi^{2}=\Pi$, where $\Omega^{1}(B)$ is the restriction of
$\Omega^{1}(A)$ to a fodc on $B$. We are interested in principal connections
because they allow us to construct affine connections for any relative Hopf
module. Again, to keep preliminaries to a minimum, we will restrict our
attention to the subcategory ${}^{A}_{B}\mathrm{Mod}_{0}$, since any simple
object $\mathcal{F}$ in this subcategory is isomorphic to a subobject of $A$
(see [6, Appendix A.3] for details). Now if the quantum principal bundle is
_strong_ , that is to say, if
$(\mathrm{id}-\Pi)\circ\mathrm{d}B\subseteq\Omega^{1}(B)A$, then an affine
connection for ${\mathcal{F}}\subseteq A$ is given by

$\displaystyle\nabla:{\mathcal{F}}\to\Omega^{1}(B)\otimes_{B}{\mathcal{F}},$
$\displaystyle f\mapsto
j\\!\left(\sum_{i}(\mathrm{id}-\Pi)(\mathrm{d}f)\right)\\!,$

where $j:\Omega^{1}(B){\mathcal{F}}\to\Omega^{1}(B)\otimes_{B}{\mathcal{F}}$
is the inverse to the obvious multiplication map from
$\Omega^{1}(B)\otimes_{B}{\mathcal{F}}$ to $\Omega^{1}(B){\mathcal{F}}$.
(Faithful flatness guarantees that the multiplication map is invertible). We
call $\nabla$ the affine connection _associated_ to $\Pi$. If $\Pi$ is
additionally assumed to be a left $A$-comodule map, then every associated
affine connections will be a left $A$-comodule map.

###  6.2. Holomorphic Structures and Principal Connections

Since the Lusztig fodc on the Hopf algebra
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is left
$\mathcal{O}(\mathbb{T}^{n})$-covariant we have a homogeneous quantum