Bimodule Connections for Relative Line Modules

Bimodule Connections for Relative Line Modules
over the Irreducible Quantum Flag Manifolds

Alessandro CAROTENUTO and Réamonn Ó BUACHALLA A. Carotenuto and R. Ó Buachalla
Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech
Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received April 14, 2022, in final form September 18, 2022; Published online
October 01, 2022

It was recently shown (by the second author and Díaz García, Krutov, Somberg,
and Strung) that every relative line module over an irreducible quantum flag
manifold $\mathcal{O}_{q}(G/L_{S})$ admits a unique
$\mathcal{O}_{q}(G)$-covariant connection with respect to the
Heckenberger–Kolb differential calculus $\Omega^{1}_{q}(G/L_{S})$. In this
paper we show that these connections are bimodule connections with an
invertible associated bimodule map. This is proved by applying general results
of Beggs and Majid, on principal connections for quantum principal bundles, to
the quantum principal bundle presentation of the Heckenberger–Kolb calculi
recently constructed by the authors and Díaz García. Explicit presentations of
the associated bimodule maps are given first in terms of generalised quantum
determinants, then in terms of the FRT presentation of the algebra
$\mathcal{O}_{q}(G)$, and finally in terms of Takeuchi’s categorical
equivalence for relative Hopf modules.

quantum groups; noncommutative geometry; bimodule connections; quantum
principal bundles; quantum flag manifolds; complex geometry

46L87; 81R60; 81R50; 17B37; 16T05

##  1 Introduction

Left connections for left modules (and right connections for right modules)
are basic structures in noncommutative geometry. They generalise connections
for a vector bundle $V$, where of course the left and right actions on the
space of sections $\Gamma(V)$ coincide. However, in the noncommutative setting
it is most convenient, where possible, to consider bimodules. This prompted
the development of a theory of bimodule connections, which is to say, left
(right) connections, together with a compatibility for the right (left)
action, described in terms of an associated bimodule map. The literature on
bimodule connections spans twenty-five years, from the original works [12, 14,
15, 23, 24] to more recent works such as [1, 2, 4, 16, 26]. The interested
reader can find a very thorough exposition of these topics in the recent
monograph [3].

Quantum principal bundles and _principal_ connections were introduced by
Brzeziński and Majid [6] as a method for constructing left connections for
modules associated to right comodule algebras $B=P^{\textrm{co}(H)}$. Now
associated modules are naturally bimodules, and so, it is natural to ask when
a principal connection induces a bimodule connection on an associated module.
This question has been thoroughly investigated by Beggs and Majid [3, Section
5.4], resulting in a number of sufficient criteria for determining when
bimodule connections are produced.

This provides us with powerful machinery to construct bimodule connections,
and this paper concerns itself with an application of this machinery to the
theory of Drinfeld–Jimbo quantum groups. Examples of bimodule connections in
the setting of Drinfeld–Jimbo quantum groups have been constructed for the
quantum plane in [13], then for the relative line modules $\mathcal{E}_{k}$
over the standard Podleś sphere in [20] and [3, Example 5.23], for the quantum
projective plane in [21], and for the first-order Heckenberger–Kolb calculi of
quantum projective space in [26]. In [11] the second author, together with
Díaz García, Krutov, Somberg and Strung, showed that the relative line modules
over all irreducible quantum flag manifolds $\mathcal{O}_{q}(G/L_{S}),$ with
the first order Heckenberger–Kolb calculus, admit a unique
$\mathcal{O}_{q}(G)$-covariant connection. The same connection was then used
when proving the Borel Weil theorem for quantum Grassmannians in [8] and for
the irreducible quantum flag manifolds in [7]. In this paper we use the theory
of quantum principal bundles to extend Beggs and Majid’s work on the standard
Podleś sphere to the setting of the irreducible quantum flag manifolds
$\mathcal{O}_{q}(G/L_{S})$, and so, produce a large family of systematically
constructed bimodule connections for Drinfeld–Jimbo quantum spaces.

In more detail, the irreducible quantum flag manifolds are a broad family of
quantum homogeneous spaces, with the standard Podleś sphere, and more
generally quantum projective space, as the most tractable examples. From the
seminal work of Heckenberger and Kolb [17, 18], we know that each
$\mathcal{O}_{q}(G/L_{S})$ possesses an essentially unique
$\mathcal{O}_{q}(G)$-covariant $q$-deformation of its classical de Rham
complex $\Omega^{\bullet}_{q}(G/L_{S})$, constituting one of the most
important structures in the noncommutative geometry of quantum groups. The
relative line modules $\mathcal{E}_{k}$ over each $\mathcal{O}_{q}(G/L_{S})$,
which are indexed by integers $k\in\mathbb{Z}$, were shown in [11] to
possesses a unique left $\mathcal{O}_{q}(G)$-covariant connection $\nabla$
with respect to $\Omega^{\bullet}_{q}(G/L_{S})$. Now as established in the
series of papers [7, 8, 9], the Heckenberger–Kolb calculi admit quantum
principal bundle descriptions. In this presentation, the total algebra of the
quantum bundle is the quantum Poisson homogeneous space

$\displaystyle\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}\simeq\bigoplus_{k\in\mathbb{Z}}\mathcal{E}_{k},$

and the first-order calculus is the Heckenberger–Kolb calculus over
$\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}$ introduced in [18,
Section 3.2.3]. The zero map was then shown to be a strong principal
connection, allowing the connection for each line module $\mathcal{E}_{k}$ to
be realised as an associated connection. Combining this result with Beggs and
Majid’s general results on bimodule connections allows us to conclude that
each $\nabla$ is a bimodule connection, as are its holomorphic and anti-
holomorphic summands. An immediate application is a description, for each
$k\in\mathbb{Z}_{>0}$, of the connections on $\mathcal{E}_{\pm k}$ as the
$k$-fold tensor product of the $\mathcal{E}_{\pm 1}$ connection.

Our proof of the bimodule property has the advantage of being simple and
conceptual, but the disadvantage of not providing an explicit description of
the associated bimodule map $\sigma$. To address this we produce three
complementary explicit descriptions of $\sigma$. The first approach uses a
family of identities, which we loosely term _generalised determinants_ ,
coming from a dual basis description of the copy of the trivial module in the
tensor product $V_{\varpi_{x}}\otimes{}^{\vee}V_{\varpi_{x}}$, where
$V_{\varpi_{x}}$ is the simple $U_{q}(\mathfrak{g})$-module corresponding to
the crossed Dynkin node, and ${}^{\vee}V_{\varpi_{x}}$ is the dual module.
This extends the foundational work done in [25] for the special case of
standard Podlés sphere. The second approach realises the generalised
determinant identities, for the $A$ and $B$-series, in terms of the well-known
FRT presentation [30] of the non-exceptional quantum coordinate algebras
$\mathcal{O}_{q}(G)$. For the $A$-series, which is to say for the quantum
Grassmannians, this gives an expression of the bimodule map in terms of the
well-known quantum minors. Finally, we use Takeuchi’s equivalence for relative
Hopf modules [32] to give an alternative description of $\sigma$ for all the
irreducible quantum flag manifolds.

We consider our results on tensor products to be an important step towards a
conceptual explanation for the phenomenon of $q$-deformed additivity of tensor
product curvature [3, Example 5.51], and we intend to investigate this in
future work.

###  1.1 Summary of the paper

The paper is organised as follows: in Section 2 we recall basic material on
first-order differential calculi, Hopf–Galois extensions, and quantum
principal bundles. Moreover, we briefly recall necessary material about
Drinfeld–Jimbo quantum groups, their quantum flag manifolds, and their
relative line modules.

In Section 3 we give a simple accompaniment to Beggs and Majid’s criteria for
a strong principal connection to induce bimodule connections. We then recall
the recent classification of $\mathcal{O}_{q}(G)$-covariant connections for
line modules over the irreducible quantum flag manifolds [11]. We also recall
the recent quantum principal bundle description of the Heckenberger–Kolb
calculi [7]. Building directly on this material, we prove the main result of
the paper:

######  Theorem 1.1.

For each line module $\mathcal{E}_{k}$, the connection $\nabla$ is a bimodule
connection. Moreover, the associated bimodule map is a left
$\mathcal{O}_{q}(G)$-comodule isomorphism.

We also make some comments on the associated tensor product connections.

In Section 4 explicit presentations of the associated bimodule maps are given
in terms of generalised quantum determinants, then in terms of the FRT
presentations of $\mathcal{O}_{q}(\mathrm{SU}_{n})$ and
$\mathcal{O}_{q}(\mathrm{SO}_{2n+1})$, and finally in terms of Takeuchi’s
categorical equivalence for relative Hopf modules.

In Appendix A we recall some useful material about the irreducible quantum
flag manifolds.

##  2 Preliminaries

In this section we briefly recall standard material on first-order
differential calculi, Hopf–Galois extensions, and quantum principal bundles.
We refer the reader to [3] for a more detailed presentation. Moreover, we
briefly recall necessary material about Drinfeld–Jimbo quantum groups, their
quantum flag manifolds, and their relative line modules.

###  2.1 First-order differential calculi and bimodule connections

In this subsection we recall the notion of a bimodule connection for a
differential calculus, see [3, Section 3.4] for further details. A first-order
differential calculus (fodc) over an algebra $B$ is a pair
$\big{(}\Omega^{1}(B),\mathrm{d}\big{)}$, where $\Omega^{1}(B)$ is a
$B$-bimodule and $\mathrm{d}\colon B\to\Omega^{1}(B)$ is a derivation such
that $\Omega^{1}$ is generated as a left $B$-module by those elements of the
form $\mathrm{d}b$, for $b\in B$. The universal fodc over $B$ is the pair
$\big{(}\Omega^{1}_{u}(B),\mathrm{d}_{u}\big{)}$, where $\Omega^{1}_{u}(B)$ is
the kernel of the multiplication map $m_{B}\colon B\otimes B\to B$, and
$\mathrm{d}_{u}\colon B\to\Omega^{1}_{u}(B)$ is defined by
$\mathrm{d}_{u}b=1\otimes b-b\otimes 1$. Every fodc over $B$ can be obtained
as the quotient of $\Omega^{1}_{u}(B)$ by a $B$-sub-bimodule
$N\subset\Omega^{1}_{u}(B)$. For any subalgebra $B^{\prime}\subseteq B$, the
_restriction_ of a fodc over $B$ to $B^{\prime}$ is the fodc
$\Omega^{1}(B^{\prime})\subseteq\Omega^{1}(B)$ over $B^{\prime}$ generated by
the elements $\mathrm{d}b$, for $b\in B^{\prime}$.

For $\Omega^{1}(B)$ a fodc over an algebra $B$ and $\mathcal{F}$ a left
$B$-module, a _left connection_ on $\mathcal{F}$ is a $\mathbb{C}$-linear map
$\nabla\colon\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$ satisfying the
_left Leibniz rule_

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

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

$\displaystyle\sigma(f\otimes\mathrm{d}b)=\nabla(fb)-\nabla(f)b.$ (2.1)

Note that in the commutative case
$\sigma(f\otimes\mathrm{d}b)=\mathrm{d}b\otimes f$, that is, $\sigma$ reduces
to the flip map.

Let $\mathcal{F}$ be a bimodule, $(\nabla_{\mathcal{F}},\sigma_{\mathcal{F}})$
a bimodule connection, and $\mathcal{E}$ a bimodule with bimodule connection
$(\nabla_{\mathcal{E}},\sigma_{\mathcal{E}})$, we define the connection for
$\mathcal{F}\otimes_{B}\mathcal{E}$ as follows

$\displaystyle\nabla_{\mathcal{F}\otimes\mathcal{E}}=\nabla_{\mathcal{F}}\otimes\mathrm{id}_{\mathcal{E}}+(\sigma_{\mathcal{F}}\otimes\mathrm{\mathrm{id}}_{\mathcal{E}})\circ(\mathrm{id}_{\mathcal{F}}\otimes\nabla_{\mathcal{E}}).$
(2.2)

Note that $\nabla_{\mathcal{F}\otimes\mathcal{E}}$ is again a bimodule
connection, with bimodule map
$(\sigma_{\mathcal{F}}\otimes\mathrm{id})\circ(\mathrm{id}\otimes\sigma_{\mathcal{E}})$.

######  Example 2.1.

For any fodc $\big{(}\Omega^{1}(B),\mathrm{d}\big{)}$, through the isomorphism
$\Omega^{1}(B)\simeq\Omega^{1}(B)\otimes_{B}B$, we can consider
$\mathrm{d}\colon B\to\Omega^{1}(B)$ as a connection for $B$. Since

$\mathrm{d}(bb^{\prime})-\mathrm{d}(b)b^{\prime}=b\mathrm{d}(b^{\prime}),$

it holds that this is a bimodule connection with associated bimodule map

$\displaystyle\sigma(b\otimes\mathrm{d}b^{\prime})=b\mathrm{d}b^{\prime}\otimes
1.$

This elementary bimodule connection will be used in the proof of Lemma 4.5
below.

Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be two left $B$-bimodules, and
let $\nabla\colon\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$ and
$\nabla^{\prime}\colon\mathcal{F}^{\prime}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}^{\prime}$
be two connections, then we say that $\nabla$ and $\nabla^{\prime}$ are
_isomorphic_ if we have an isomorphism of $B$-bimodules
$\alpha\colon\mathcal{F}\to\mathcal{F}^{\prime}$ such that the following
diagram commutes:

$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\nabla}$$\textstyle{\Omega^{1}(B)\otimes_{B}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\otimes\alpha}$$\textstyle{\mathcal{F}^{\prime}\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla^{\prime}}$$\textstyle{\Omega^{1}(B)\otimes_{B}\mathcal{F}^{\prime}.}$

###  2.2 Quantum principal bundles

Let $(P,\Delta_{R})$ be a right $H$-comodule algebra, for some Hopf algebra
$H$. A fodc over $P$ is said to be right $H$-covariant if the following
(necessarily unique) map is well defined

$\displaystyle\Delta_{R}\colon\ \Omega^{1}(P)\to\Omega^{1}(P)\otimes H,\qquad
p\mathrm{d}q\mapsto p_{(0)}\mathrm{d}q_{(0)}\otimes p_{(1)}q_{(1)}.$

Covariance for a fodc over a left comodule algebra is defined similarly.

Let $(P,\Delta_{R})$ be a right $H$-comodule algebra. We say that $P$ is a
Hopf–Galois extension of $B=P^{\,\mathrm{co}(H)}$ a bijection is given by

$\mathrm{can}:=(m_{P}\otimes\mathrm{id})\circ(\mathrm{id}\otimes\Delta_{R})\colon\
P\otimes_{B}P\to P\otimes H,$

where $m_{P}$ denotes the multiplication of $P$. A _right $H$-principal
comodule algebra_ is a right $H$-comodule algebra $P$, such that $P$ is a
Hopf–Galois extension of $B$, and such that $P$ is faithfully flat as a
$B$-bimodule.

######  Definition 2.2.

Let $H$ be a Hopf algebra. A quantum principal $H$-bundle is a pair
$\big{(}P,\Omega^{1}(P)\big{)}$, consisting of a right $H$-principal comodule
algebra $(P,\Delta_{R})$, and a right $H$-covariant calculus $\Omega^{1}(P)$,
such that, if $N\subseteq\Omega^{1}_{u}(P)$ is the sub-bimodule of the
universal calculus corresponding to $\Omega^{1}(P)$, we have
$\mathrm{ver}(N)=P\otimes I$, for some $\mathrm{Ad}$-subcomodule right ideal

$I\subseteq H^{+}:=\ker(\varepsilon\colon H\to\mathbb{C}),$

where $\mathrm{Ad}\colon H\to H\otimes H$ is defined by
$\mathrm{Ad}(h):=h_{(2)}\otimes S(h_{(1)})h_{(3)}$. A quantum principal bundle
is called strong when it holds that $\Omega^{1}(B)P=P\Omega^{1}(B)$.

We can now define a principal connection for a quantum principal bundle. In
Section 2.3 below we will show how to define connections for a certain class
of noncommuative vector bundles starting from a principal connection.

######  Definition 2.3.

A principal connection for a quantum principal $H$-bundle
$\big{(}P,\Omega^{1}(P)\big{)}$ is a left $P$-module, right $H$-comodule,
projection $\Pi\colon\Omega^{1}(P)\to\Omega^{1}(P)$ satisfying

$\ker(\Pi)=P\Omega^{1}(B)P.$

A principal connection $\Pi$ is called strong if
$(\mathrm{id}-\Pi)\big{(}\mathrm{d}P\big{)}\subseteq\Omega^{1}(B)P$. It is
clear that for any strong principal bundle, all principal connections are
strong.

###  2.3 Connections from principal connections

For $P$ a right principal $H$-comodule algebra, we denote by
${}_{P}\mathrm{Mod}$ the category of left $P$-modules, and we define a functor

$\displaystyle\Psi\colon\ {}^{H}\mathrm{Mod}\to{}_{B}\mathrm{Mod}_{B},\qquad
V\mapsto P\,\square_{H}V,$

where the right $B$-action is given by multiplication on the first tensor
factor, and that acts on morphisms $\gamma\colon V\to W$ as
$\Psi(\gamma)=\mathrm{id}\otimes\gamma$. We call any left $P$-module of the
form $\Psi(V)$, for some $V\in{}^{H}\mathrm{Mod}$, an _associated module_. We
can use principal connections to define a connection
$\nabla\colon\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$ for any left
$P$-module of the form $\mathcal{F}:=\Psi(V).$ Note first that we have a
natural embedding

$\displaystyle j\colon\
\Omega^{1}(B)\otimes_{B}\mathcal{F}\hookrightarrow\Omega^{1}(B)P\,\square_{H}V$

given by the multiplication map. A strong principal connection $\Pi$
determines a connection $\nabla$ on $\mathcal{F}$ by

$\displaystyle\nabla\colon\
\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F},\qquad\sum_{i}p_{i}\otimes
v_{i}\mapsto j^{-1}\big{(}(\mathrm{id}-\Pi)(\mathrm{d}p_{i})\otimes
v_{i}\big{)}.$

Indeed, since $\mathrm{d}$ and the projection $\Pi$ are both right
$H$-comodule maps, their composition
$(\mathrm{id}-\Pi)\allowbreak\circ\mathrm{d}\otimes\mathrm{id}$ is a right
$H$-comodule map. Hence the image of
$(\mathrm{id}-\Pi)\circ\mathrm{d}\otimes\mathrm{id}_{V}$ is contained in
$j\big{(}\Omega^{1}(B)\otimes_{B}\mathcal{F}\big{)}$.

Let us now additionally assume that $P$ admits a left $A$-coaction giving it
the structure of an $(A,H)$-comodule algebra, that $\Omega^{1}(P)$ is also
left $A$-covariant, and that $\Pi$ is also a left $A$-comodule map. In this
case, any $\mathcal{F}$ of the form $P\,\square_{H}V$ has an obvious left
$A$-comodule structure, and the associated connection $\nabla$ is a left
$A$-comodule map.

###  2.4 Drinfeld–Jimbo quantum groups

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra of
rank $l$. We fix a Cartan subalgebra $\mathfrak{h}$ and choose a set of simple
roots $\Pi=\\{\alpha_{1},\dots,\alpha_{l}\\}$ for the corresponding root
system. Denote by $(\cdot,\cdot)$ the symmetric bilinear form induced on
$\mathfrak{h}^{*}$ by the Killing form of $\mathfrak{g}$, normalised so that
any shortest simple root $\alpha_{i}$ satisfies $(\alpha_{i},\alpha_{i})=2$.
The Cartan matrix $(a_{ij})$ of $\mathfrak{g}$ is defined by
$a_{ij}:=\big{(}\alpha_{i}^{\vee},\alpha_{j}\big{)}$, where
$\alpha_{i}^{\vee}:=2\alpha_{i}/(\alpha_{i},\alpha_{i})$. The set of
fundamental weights $\\{\varpi_{1},\dots,\varpi_{l}\\}$ of $\mathfrak{g}$ is
the dual basis of simple coroots
$\big{\\{}\alpha_{1}^{\vee},\dots,\alpha_{l}^{\vee}\big{\\}}$. We denote by
$\mathcal{P}$ the $\mathbb{Z}$-span of the fundamental weights, and by
${\mathcal{P}}^{+}$ the $\mathbb{Z}_{\geq 0}$-span of the fundamental weights.

Let $q\in\mathbb{R}$ such that $q\neq-1,0,1$, and denote
$q_{i}:=q^{(\alpha_{i},\alpha_{i})/2}$. The Drinfeld–Jimbo quantised
enveloping algebra $U_{q}(\mathfrak{g})$ is the noncommutative associative
algebra generated by the elements $E_{i}$, $F_{i}$, $K_{i}$, and $K^{-1}_{i}$,
for $i=1,\ldots,l$, subject to the relations

$\displaystyle K_{i}E_{j}=q_{i}^{a_{ij}}E_{j}K_{i},\qquad
K_{i}F_{j}=q_{i}^{-a_{ij}}F_{j}K_{i},\qquad K_{i}K_{j}=K_{j}K_{i},\qquad
K_{i}K_{i}^{-1}=K_{i}^{-1}K_{i}=1,$ $\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{K_{i}-K^{-1}_{i}}{q_{i}-q_{i}^{-1}},$

and the quantum Serre relations which we omit (see [22, Section 6.1.2] for
details). The formulae

$\displaystyle\Delta(K_{i})=K_{i}\otimes
K_{i},\qquad\Delta(E_{i})=E_{i}\otimes K_{i}+1\otimes
E_{i},\qquad\Delta(F_{i})=F_{i}\otimes 1+K_{i}^{-1}\otimes F_{i}$

define a Hopf algebra structure on $U_{q}(\mathfrak{g})$, satisfying
$\varepsilon(E_{i})=\varepsilon(F_{i})=0$, and $\varepsilon(K_{i})=1$.

For any $U_{q}(\mathfrak{g})$-module $V$, a vector $v\in V$ is called a
_weight vector_ of _weight_ $\mathrm{wt}(v)\in\mathcal{P}$ if
$K_{i}\triangleright v=q^{(\alpha_{i},\mathrm{wt}(v))}v$, for all
$i=1,\ldots,r$. For each $\lambda\in\mathcal{P}^{+}$ there exists an
irreducible finite-dimensional $U_{q}(\mathfrak{g})$-module $V_{\lambda}$,
uniquely defined by the existence of a weight vector $v_{\mathrm{hw}}\in
V_{\lambda}$ with weight $\lambda$, called a highest weight vector, satisfying
$E_{i}\triangleright v_{\mathrm{hw}}=0$, for all $i=1,\dots,l$. We call any
finite direct sum of such $U_{q}(\mathfrak{g})$-representations a type-$1$
module. Each type-$1$ module is spanned by weight vectors.

Let $V$ be a finite-dimensional $U_{q}(\mathfrak{g})$-module, $v\in V$, and
$f\in V^{*}$, the linear dual of $V$. Consider the function

$\displaystyle c^{\textrm{\tiny$V$}}_{f,v}\colon\
U_{q}(\mathfrak{g})\to\mathbb{C},\qquad X\mapsto f\big{(}X(v)\big{)}.$

Consider now the Hopf subalgebra of $U_{q}(\mathfrak{g})^{\circ}$, the Hopf
dual of $U_{q}(\mathfrak{g})$, generated by all functions of the form
$c^{\textrm{\tiny$V$}}_{f,v}$, for $V$ a type-$1$ representation. We denote
this Hopf algebra by $\mathcal{O}_{q}(G)$ and call it the Drinfeld–Jimbo
quantum coordinate algebra of $G$, where $G$ is the compact, simply-connected,
simple Lie group having $\mathfrak{g}$ as its complexified Lie algebra.

###  2.5 Quantum flag manifolds

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

$\displaystyle
U_{q}(\mathfrak{l}_{S}):=\big{\langle}K_{i},E_{s},F_{s}\,|\,i=1,\ldots,l=\mathrm{rank}(\mathfrak{g});\,s\in
S\big{\rangle}\subseteq U_{q}(\mathfrak{g}).$

The left action of $U_{q}(\mathfrak{g})$ on $\mathcal{O}_{q}(G)$ restricts to
a left $U_{q}(\mathfrak{l}_{S})$-action. We call the invariant subspace

$\displaystyle\mathcal{O}_{q}(G/L_{S}):={}^{U_{q}(\mathfrak{l}_{S})}\mathcal{O}_{q}(G),$

the _quantum flag manifold_ associated to $S$. We note that
$\mathcal{O}_{q}(G/L_{S})$ is a left $\mathcal{O}_{q}(G)$-comodule algebra by
construction. Moreover, $\mathcal{O}_{q}(G)$ is faithfully flat as a right
$\mathcal{O}_{q}(G/L_{S})$-module (see for example [9, Section 5.4]). It now
follows from [32, Theorem 1] that $\mathcal{O}_{q}(G/L_{S})$ coincides with
the space of right coinvariants of the coaction
$\Delta_{R}:=(\mathrm{id}\otimes\pi_{S})\circ\Delta$, where

$\pi_{S}\colon\
\mathcal{O}_{q}(G)\to\mathcal{O}_{q}(L_{S}):=\mathcal{O}_{q}(G)/\mathcal{O}_{q}(G/L_{S})^{+}\mathcal{O}_{q}(G)$

is the canonical Hopf algebra projection, with
$\mathcal{O}_{q}(G/L_{S})^{+}:=\mathcal{O}_{q}(G/L_{S})\cap\ker(\varepsilon)$.

Similarly we can define the Hopf subalgebra

$\displaystyle
U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S}):=\big{<}K_{s},E_{s},F_{s}\,|\,s\in
S\big{>}\subseteq U_{q}(\mathfrak{l}_{S})\subseteq U_{q}(\mathfrak{g}).$

We call the invariant subspace

$\displaystyle\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}):={}^{U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S})}\mathcal{O}_{q}(G)$

the _quantum Poisson homogeneous space_ associated to $S$.

###  2.6 Relative line modules

From here on, we will restrict our attention to the irreducible quantum flag
manifolds $\mathcal{O}_{q}(G/L_{S})$ (see Appendix A for the definition of
irreducible). In this case $\mathcal{O}_{q}(L_{S})$ contains a Hopf subalgebra
isomorphic to $\mathcal{O}(U_{1})$, the Hopf algebra of Laurent polynomials.
Moreover, the left $\mathcal{O}_{q}(L_{S})$-coaction on $\mathcal{O}_{q}(G)$
restricts to an $\mathcal{O}(U_{1})$-coaction on
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$, which is to say

$\Delta_{R}(\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}))\subseteq\mathcal{O}_{q}(G)\otimes\mathcal{O}(U_{1}).$

This gives $\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ the structure of a right
$\mathcal{O}(U_{1})$-comodule algebra. We denote the associated
$\mathbb{Z}$-algebra grading by

$\displaystyle\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})=:\bigoplus_{k\in\mathbb{Z}}\mathcal{E}_{k}.$
(2.3)

This grading is strong, as explained in [9, Section 4], meaning that each
$\mathcal{E}_{k}$ is a line module in the sense of [3, Section 3.5]. Moreover,
we see that each $\mathcal{E}_{k}$ admits a natural presentation as an
associated module to our $\mathcal{O}(U_{1})$-comodule algebra (see [9,
Section 5] for a detailed discussion).

By construction each $\mathcal{E}_{k}$ is a left
$\mathcal{O}_{q}(G)$-subcomodule of $\mathcal{O}_{q}(G)$, and so, it is a
relative Hopf module in the sense of Takeuchi (see Section 4.3). Hence,
following [10, Section 3.1] and [9, Section 2.3] we refer to each
$\mathcal{E}_{k}$ as a _relative line module_.

We can give an explicit description of this grading as follows: let
$\alpha_{x}$ be the simple root not contained in $S$, and let $\varpi_{x}$ be
the corresponding simple weight. Choose a weight basis $\\{v_{i}\\}_{i=1}^{N}$
of $V_{\varpi_{x}}$ such that $v_{N}$ is a highest weight vector. For $w_{0}$
the longest element of the Weyl group of $\mathfrak{g}$, we denote the dual
basis of $V_{-w_{0}(\varpi_{x})}$ by $\\{f_{i}\\}_{i=1}^{N}$. From [31,
Theorem 4.1], we know that a set of generators for
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ is given by

$\displaystyle
z_{i}:=c^{V_{\varpi_{x}}}_{f_{i},v_{N}},\qquad\overline{z}_{j}:=c^{V_{-w_{0}(\varpi_{x})}}_{v_{j},f_{N}}\qquad\text{for
}i,j=1,\dots,N:=\dim(V_{\varpi_{x}}).$

The $\mathbb{Z}$-grading on
$\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}$ is completely determined
by $\deg(z_{1})=1$, and $\deg(\overline{z}_{i})=-1$, for all $i=1,\dots,N$.

##  3 Bimodule connections for the relative line modules

In this section we give a simple accompaniment to Beggs and Majid’s criteria
for a strong principal connection to induce bimodule connections. We then
recall the recent classification of $\mathcal{O}_{q}(G)$-covariant connections
$\nabla\colon\mathcal{E}_{k}\to\Omega^{1}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{k}$.
We also recall the recent quantum principal bundle description of the
Heckenberger–Kolb calculi [7]. Building directly on this material, we then
prove the main result of the paper, namely that $\nabla$, and its holomorphic
and anti-holomorphic summands, are bimodule connections. We also make some
comments on the associated tensor product connections.

###  3.1 Bimodule principal connections

The question of when a principal connection determines a bimodule connection
was thoroughly examined by Beggs and Majid in [3, Section 5.4] and [3,
Proposition 5.54], and sufficient conditions were given for a strong principal
connection $\Pi$ to induce a bimodule connection for any associated module. We
now prove a simple accompaniment to these results, suited to our needs below.
It takes a weaker set of assumptions and allows for an elementary self-
contained proof.

######  Theorem 3.1.

Let $\big{(}P,\Omega^{1}(P)\big{)}$ be a quantum principal bundle, endowed
with a strong principal connection $\Pi\colon\Omega^{1}(P)\to\Omega^{1}(P)$
that is additionally assumed to be a right $B$-module map.

  1. $1.$

The associated connection
$\nabla\colon\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$, for each
associated module $\mathcal{F}$, is a bimodule connection.

  2. $2.$

The associated bimodule map
$\sigma\colon\mathcal{F}\otimes_{B}\Omega^{1}(B)\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$
is an injective right $H$-comodule map. Moreover, if
$\big{(}P,\Omega^{1}(P)\big{)}$ is strong then $\sigma$ is an isomorphism.

  3. $3.$

Assume that $P$ is endowed with a left $A$-coaction making it a left
$A$-comodule algebra, and moreover, assume that $\Omega^{1}(P)$ is a left
$A$-covariant fodc. If $\Pi$ is a left $A$-comodule map, then $\sigma$ is a
left $A$-comodule map.

###### Proof.

1\. Writing $\mathcal{F}$ as $P\square_{H}V$, for $V\in{}^{H}\mathrm{Mod},$ we
let $f=\sum_{i}p_{i}\otimes v_{i}\in\mathcal{F}$ and $\omega\in\Omega^{1}(B)$,
we claim that a $B$-bimodule map
$\sigma\colon\mathcal{F}\otimes_{B}\Omega^{1}(B)\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$
is given by

$\displaystyle\sigma\left(f\otimes\omega\right)=j^{-1}\bigg{(}\sum_{i}p_{i}\omega\otimes
v_{i}\bigg{)}.$

Indeed, since $\Pi$ is a strong connection, we have that

$\displaystyle\sum_{i}p_{i}\omega=\sum_{i}(\mathrm{id}-\Pi)p_{i}\omega\in\Omega^{1}(B)P,$

and so, the map $\sigma$ is well defined. Now we prove that $(\nabla,\sigma)$
is a bimodule connection by showing that expression (2.1) holds. We have the
following:

$\displaystyle\nabla(fb)-\nabla(f)b$
$\displaystyle=\sum_{i}j^{-1}\big{(}(\mathrm{id}-\Pi)\mathrm{d}(p_{i}b)\otimes
v_{i}\big{)}-\sum_{i}j^{-1}\big{(}(\mathrm{id}-\Pi)\mathrm{d}p_{i}\otimes
v_{i}\big{)}b$
$\displaystyle=\sum_{i}j^{-1}\big{(}(\mathrm{id}-\Pi)\mathrm{d}(p_{i}b)\otimes
v_{i}\big{)}-\sum_{i}j^{-1}\big{(}(\mathrm{id}-\Pi)\mathrm{d}(p_{i})b\otimes
v_{i}\big{)}.$

By the Leibniz rule this reduces to

$\displaystyle\sum_{i}j^{-1}\big{(}(\mathrm{id}-\Pi)p_{i}\mathrm{d}b\otimes
v_{i}\big{)}=\sum_{i}j^{-1}\big{(}p_{i}\mathrm{d}b\otimes
v_{i}\big{)}=\sigma(f\otimes\mathrm{d}b),$

where the last identity follows from the fact that each summand
$p_{i}\mathrm{d}b$ is an horizontal form.

2\. It is clear from the above formula that $\sigma$ is injective. Covariance
of the fodc and the fact that both $j^{-1}$ and $\Pi$ are left $H$-comodule
maps, imply that $\sigma$ is a right $H$-comodule map. If
$\big{(}P,\Omega^{1}(P)\big{)}$ is strong, then we have an isomorphism

$\displaystyle\mu^{\prime}\colon\ \Omega^{1}(B)\otimes_{B}P\to P\Omega^{1}(B)$

given by multiplication in $\Omega^{1}(P)$. We have an additional natural
embedding

$\displaystyle j^{\prime}\colon\
\mathcal{F}\otimes_{B}\Omega^{1}(B)\hookrightarrow
P\Omega^{1}(B)\,\square_{H}V,$

induced by the multiplication map. The composition of isomorphisms
$j^{\prime-1}\circ\mu$ then gives an isomorphism between
$\Omega^{1}(B)\otimes_{B}\mathcal{F}$ and
$\mathcal{F}\otimes_{B}\Omega^{1}(B)$, which we see is an inverse for
$\sigma$.

3\. In this case, as discussed in Section 2.3, the associated connection
$\nabla$ will be a left $A$-comodule map. Hence it follows directly from the
definition of $\sigma$ that it will also be a left $A$-comodule map. ∎

###  3.2 Relative line module connections for the Heckenberger–Kolb calculi

As established in the seminal paper [18], there exist, up to isomorphism,
exactly two finite-dimensional irreducible left $\mathcal{O}_{q}(G)$-covariant
fodc over every irreducible quantum flag manifold. We denote these calculi by

$\displaystyle\big{(}\Omega^{(1,0)}_{q}(G/L_{S}),\partial\big{)},\qquad\big{(}\Omega^{(0,1)}_{q}(G/L_{S}),\overline{\partial}\big{)}.$

Moreover, we call their direct sum the _Heckenberger–Kolb calculus_ on
$\mathcal{O}_{q}(G/L_{S})$ and we denote it by

$\big{(}\Omega^{1}_{q}(G/L_{S}),\mathrm{d}:=\overline{\partial}+\partial\big{)}.$

The task of describing the left $\mathcal{O}_{q}(G)$-covariant connections
$\nabla\colon\mathcal{E}_{k}\to\Omega^{1}_{q}(G/L_{S})\otimes\mathcal{E}_{k}$
was recently undertaken by the second author and Díaz García, Krutov, Somberg,
and Strung. The proof of the following result uses the theory of principal
comodule algebras [5] to a construct universal connection, and then quotients
to the special case of the Heckenberger–Kolb calculi.

######  Theorem 3.2 ([11, Theorem 4.5]).

Let $\mathcal{O}_{q}(G/L_{S})$ be an irreducible quantum flag manifold, then
for every $k\in\mathbb{Z}$, the relative line module $\mathcal{E}_{k}$ admits
a unique $\mathcal{O}_{q}(G)$-covariant connection
$\nabla\colon\mathcal{E}_{k}\to\Omega^{1}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{k}$.

In addition to the connection $\nabla$, we can also consider the _holomorphic_
, and respectively _anti-holomorphic, connections_

$\displaystyle\partial_{\mathcal{E}_{k}}:=\big{(}\mathrm{proj}^{(1,0)}\otimes\mathrm{id}\big{)}\circ\nabla\qquad\mathrm{and}\qquad\overline{\partial}_{\mathcal{E}_{k}}:=\big{(}\mathrm{proj}^{(0,1)}\otimes\mathrm{id}\big{)}\circ\nabla,$

where $\mathrm{proj}^{(1,0)}$ denotes the projection onto
$\Omega^{(1,0)}_{q}(G/L_{S})$, and similarly for $\mathrm{proj}^{(0,1)}$. It
clearly follows from Theorem 3.2 that $\partial_{\mathcal{E}_{k}}$, and
$\overline{\partial}_{\mathcal{E}_{k}}$, are the unique covariant connections
for $\mathcal{E}_{k}$ with respect to the fodc $\Omega^{(1,0)}_{q}(G/L_{S})$,
and respectively $\Omega^{(0,1)}_{q}(G/L_{S})$.

###  3.3 A quantum principal bundle

In [18], with a view to calculating higher order relations, the fodc
$\Omega^{(1,0)}_{q}(G/L_{S})$ was obtained as the restriction of a left
$\mathcal{O}_{q}(G)$-covariant fodc over the quantum Poisson homogeneous space
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$. The explicit construction involved a
detailed $R$-matrix argument, and we refer the interested reader to [7] for a
summary in the notation of this paper. The following theorem summarises the
details relevant to this paper. The statement of the theorem makes use of
$S_{q}[G/L_{S}]$, the _quantum homogeneous coordinate ring_ of $G/L_{S}$,
which is to say, the sub-algebra of $\mathcal{O}_{q}(G)$ generated by the
elements $z_{i}$, for $i=1,\dots,N$.

######  Theorem 3.3 ([18, Section 3.2]).

There exists a left $\mathcal{O}_{q}(G)$-covariant fodc
$\Omega^{(0,1)}_{q}\big{(}G/L^{\mathrm{s}}_{S},\overline{\partial}\big{)}$
such that

  1. $(1)$

$\overline{\partial}p=0$, for all $p\in S_{q}[G/L_{S}]$,

  2. $(2)$

the fodc is right $\mathcal{O}(U_{1})$-covariant,

  3. $(3)$

the pair
$\big{(}\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)},\Omega^{(0,1)}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)}\big{)}$
is a quantum principal bundle,

  4. $(4)$

the zero map is a strong principal connection for the bundle.

The first point of the theorem follows from the construction of the fodc given
in [18, Section 3.2]. The right $\mathcal{O}(U_{1})$-covariance was
established by the authors and Díaz García in [7, Lemma 5.5]. The quantum
principal bundle property was also established by the authors and Díaz García
in [7, Proposition 5.6], where the general theory of principal pairs (as
introduced in [8, 9]) was used. An analogous presentation of
$\Omega^{(0,1)}_{q}(G/L_{S})$ was also given in [18, Section 3.2], and the
right $\mathcal{O}(U_{1})$-covariance, and the quantum principal bundle
property also holds for this fodc.

###  3.4 The bimodule property

In this section we observe that the recent results collected in the
subsections above are enough to imply that the line module connections are
bimodule connections. We also show that the associated bimodule maps are
invertible, we necessitates the following preliminary lemma.

######  Lemma 3.4.

The quantum principal bundles

$\displaystyle\big{(}\mathcal{O}_{q}\big{(}G/L_{S}^{\mathrm{s}}\big{)},\Omega^{(0,1)}_{q}\big{(}G/L_{S}^{\mathrm{s}}\big{)}\big{)},\qquad\big{(}\mathcal{O}_{q}\big{(}G/L_{S}^{\mathrm{s}}\big{)},\Omega^{(1,0)}_{q}\big{(}G/L_{S}^{\mathrm{s}}\big{)}\big{)},$

are strong.

###### Proof.

It was shown in [7, Lemma 5.7] that

$\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)}\Omega^{(0,1)}_{q}(G/L_{S})\subseteq\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)}.$

We prove this lemma by showing that an analogous argument establishes the
opposite inclusion.

An arbitrary form in $\Omega^{(0,1)}_{q}(G/L_{S})$ is a sum of elements of the
form $b^{\prime}\,\overline{\partial}b$, for some
$b^{\prime},b\in\mathcal{O}_{q}(G/L_{S})$. Consider now the form

$\displaystyle
b^{\prime}\big{(}\overline{\partial}b\big{)}z_{i}\in\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)},\qquad\textrm{for
any }i=1,\dots,N.$

It follows from the observations of Section 4.1 that there exist elements
$f_{j}^{\prime}\in S_{q}[G/L_{S}]$ and $v_{j}^{\prime}\in
S_{q}\big{[}G/L^{\mathrm{op}}_{S}\big{]}$ such that
$\sum_{j=1}^{N}v_{j}^{\prime}f_{j}^{\prime}=1$. Then it follows from the
Leibniz rule that

$\displaystyle
b^{\prime}\big{(}\overline{\partial}b\big{)}\overline{z}_{i}=b^{\prime}\overline{\partial}(b\overline{z}_{i})-b^{\prime}b\overline{\partial}(\overline{z}_{i})=\sum_{j=1}^{N}b^{\prime}v^{\prime}_{j}f^{\prime}_{j}\overline{\partial}(b\overline{z}_{i})-\sum_{j=1}^{N}b^{\prime}bv^{\prime}_{j}f^{\prime}_{j}\overline{\partial}(\overline{z}_{i}).$

Moreover, since $\overline{\partial}f_{j}=0$, we have that

$\displaystyle
b^{\prime}\big{(}\overline{\partial}b\big{)}\overline{z}_{i}=\sum_{j=1}^{N}b^{\prime}v^{\prime}_{j}\overline{\partial}(f^{\prime}_{j}b\overline{z}_{i})-\sum_{j=1}^{N}b^{\prime}bv^{\prime}_{j}\overline{\partial}(f^{\prime}_{j}\overline{z}_{i}).$

Thus we see that $b^{\prime}(\overline{\partial}b)\overline{z}_{i}$ is an
element of
$\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)}\Omega^{(0,1)}_{q}(G/L_{S})$,
for any $i=1,\dots,N$. Thus we can conclude that

$\displaystyle\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)}\Omega^{(0,1)}_{q}(G/L_{S})\subseteq\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}\big{(}G/L^{\mathrm{s}}_{S}\big{)},$

and hence that the quantum principal bundle is strong. ∎

######  Theorem 3.5.

For each line module $\mathcal{E}_{k}$, the connections
$\nabla,\partial_{\mathcal{E}_{k}}$ and
$\overline{\partial}_{\mathcal{E}_{k}}$ are bimodule connections. Moreover,
the associated bimodule maps are left $\mathcal{O}_{q}(G)$-comodule
isomorphisms.

###### Proof.

Recall from Section 3.3 that the zero map on
$\Omega^{1}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}$ is a strong principal
connection that realises the line modules connections
$\overline{\partial}_{\mathcal{E}_{k}}$ as associated connections. Since the
zero map is clearly a right $\mathcal{O}_{q}(G/L_{S})$-module map, we can
conclude from Theorem 3.1 that each $\overline{\partial}_{\mathcal{E}_{k}}$ is
a bimodule connection. Moreover, since the zero map is clearly a right
$\mathcal{O}_{q}(G)$-comodule map, and by the above lemma the quantum
principal bundle is strong, we can conclude from Theorem 3.1 that the
associated bimodule map $\sigma_{k}$ is a left $\mathcal{O}_{q}(G)$-comodule
isomorphism.

A similar argument establishes that the connection
$\partial_{\mathcal{E}_{k}}$ is a bimodule conection with an invertible
bimodule map. Moreover, these two results together imply that $\nabla$ is a
bimodule connection with an invertible bimodule map. ∎

######  Remark 3.6.

The decomposition of the Heckenberger–Kolb calculus $\Omega^{1}_{q}(G/L_{S})$
into its $(1,0)$ and $(0,1)$ summands extends to a complex structure
$\Omega^{(\bullet,\bullet)}$ on the maximal prolongation of
$\Omega^{1}_{q}(G/L_{S})$. Moreover, as shown in [11, Theorem 4.5], with
respect to the complex structure $\Omega^{(\bullet,\bullet)}$ the
$(0,1)$-connections $\overline{\partial}_{\mathcal{E}_{k}}$ extend to
holomorphic structures for the line bundles $\mathcal{E}_{k}$. Thus we see
that each
$\big{(}\mathcal{E}_{k},\overline{\partial}_{\mathcal{E}_{k}}\big{)}$ is a
_holomorphic bimodule_ in the sense of Beggs and Majid [3, Definition 7.14].

###  3.5 Tensor products of connections

For $k\in\mathbb{Z}_{>0}$, let $\mathcal{E}_{k}$ be the corresponding line
module over the irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$.
Since the $\mathbb{Z}$-grading given in (2.3) is strong, it follows from [27,
Corollary 3.1], that multiplication gives an isomorphism

$\displaystyle\mathcal{E}_{k}\simeq\underbrace{\mathcal{E}_{1}\otimes_{\mathcal{O}_{q}(G/L_{S})}\cdots\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{1}}_{k\text{-times}},$

which is of course a left $\mathcal{O}_{q}(G)$-comodule map. Since the
connection $\nabla$ on $\mathcal{E}_{1}$ is a bimodule connection, we can
build a bimodule connection for $\mathcal{E}_{k}$ through repeated iterations
of formula (2.2).

######  Proposition 3.7.

The connections

  1. $(1)$

$\overline{\partial}_{\mathcal{E}_{k+l}}\colon\mathcal{E}_{k+l}\to\Omega^{(0,1)}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{k+l}$,

  2. $(2)$

$\overline{\partial}_{\mathcal{E}_{k}\otimes\mathcal{E}_{l}}\colon\mathcal{E}_{k}\otimes\mathcal{E}_{l}\to\Omega^{(0,1)}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{k}\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{l}$

are isomorphic. Moreover, the corresponding holomorphic connections are
isomorphic.

###### Proof.

Since the associated bimodule map $\sigma_{1}$ is a left
$\mathcal{O}_{q}(G)$-comodule map, the tensor product connection for
$\mathcal{E}_{k}\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{l}$ is left
$\mathcal{O}_{q}(G)$-covariant. Theorem 3.2 implies that this connection must
be isomorphic to $\overline{\partial}_{\mathcal{E}_{k+l}}$. An analogous
construction exists for the line modules $\mathcal{E}_{-k}$, for all
$k\in\mathbb{Z}$, and for the holomorphic connections
$\partial_{\mathcal{E}_{k+l}}$ and
$\partial_{\mathcal{E}_{k}\otimes\mathcal{E}_{l}}$. ∎

##  4 Some explicit descriptions of the associated bimodule maps

In this section we give three complementary presentations of the bimodule map
associated to the connection $\overline{\partial}_{\mathcal{E}_{k}}$, first in
terms of generalised quantum determinants, then in terms of the FRT
presentation of $\mathcal{O}_{q}(G)$, and finally in terms of Takeuchi’s
categorical equivalence for relative Hopf modules.

###  4.1 Generalised determinant relations

The elements $z_{N}^{k}$, and $\overline{z}_{N}^{k}$, are highest weight
vectors of weight $k\varpi_{s}$, and $-kw_{0}(\varpi_{s})$. Thus we have a
dual pair of irreducible $U_{q}(\mathfrak{g})$-modules,

$\displaystyle
V_{k}:=U_{q}(\mathfrak{g})z^{k}_{N},\qquad{}^{\vee}V_{k}:=U_{q}(\mathfrak{g})\overline{z}_{N}^{k}.$

Since both $V_{k}\otimes V_{k}^{\vee}$ and ${}^{\vee}V_{k}\otimes V_{k}$
contain a copy of the trivial module, we must have a family of elements

$\displaystyle\\{f_{i},\,f^{\prime}_{i}\\}_{i=1}^{N}\subseteq
V_{k},\qquad\\{v_{i},\,v^{\prime}_{i}\\}_{i=1}^{N}\subseteq{}^{\vee}V_{k},$

satisfying

$\displaystyle\sum_{i=1}^{N}f_{i}v_{i}=1,\qquad\sum_{i=1}^{N}v^{\prime}_{i}f^{\prime}_{i}=1.$
(4.1)

It follows from the construction of the Heckenberger–Kolb calculus
$\Omega^{(0,1)}_{q}(G/L_{S}^{\,\mathrm{s}})$ that

$\displaystyle\overline{\partial}f_{i}=\overline{\partial}f^{\prime}_{i}=0,\qquad\textrm{for
all }i=1,\dots,N,$ (4.2)

a fact that will be used in the proof of the proposition below. (Indeed this
fact is a crucial ingredient in the proof of the noncommutative Borel–Weil
theorem for $\mathcal{O}_{q}(G/L_{S})$ given in [7].)

######  Proposition 4.1.

For $k\in\mathbb{Z}_{>0}$, it holds that

  1. $1.$

The bimodule map

$\sigma_{k}\colon\
\mathcal{E}_{k}\otimes_{\mathcal{O}_{q}(G/L_{S})}\Omega^{(0,1)}_{q}(G/L_{S})\to\Omega^{(0,1)}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{k},$

associated to the connection $\overline{\partial}_{\mathcal{E}_{k}}$,
satisfies

$\displaystyle\sigma_{k}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{i=1}^{N}\overline{\partial}(ebv^{\prime}_{i})\otimes
f^{\prime}_{i}-\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})\otimes
f^{\prime}_{i}b.$

Moreover, the inverse map $\sigma^{-1}$ satisfies

$\displaystyle\sigma^{-1}_{k}\big{(}\overline{\partial}b\otimes
e\big{)}=\sum_{i=1}^{N}f_{i}\otimes\overline{\partial}(v_{i}be)-\sum_{i=1}^{N}bf_{i}\otimes\overline{\partial}(v_{i}e),$

  2. $2.$

The bimodule map

$\sigma_{-k}\colon\
\mathcal{E}_{-k}\otimes_{\mathcal{O}_{q}(G/L_{S})}\Omega^{(0,1)}_{q}(G/L_{S})\to\Omega^{(0,1)}_{q}(G/L_{S})\otimes_{\mathcal{O}_{q}(G/L_{S})}\mathcal{E}_{-k},$

associated to $\overline{\partial}_{\mathcal{E}_{-k}}$, satisfies

$\displaystyle\sigma_{-k}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{i=1}^{N}\overline{\partial}(ebf_{i})\otimes
v_{i}-\sum_{i=1}^{N}\overline{\partial}(ef_{i})\otimes v_{i}b.$

Moreover, the inverse map $\sigma^{-1}_{-k}$ satisfies

$\displaystyle\sigma^{-1}_{-k}\big{(}\overline{\partial}b\otimes
e\big{)}=\sum_{i=1}^{N}v^{\prime}_{i}\otimes\overline{\partial}(f^{\prime}_{i}be)-\sum_{i=1}^{N}bv^{\prime}_{i}\otimes\overline{\partial}(f^{\prime}_{i}e).$

###### Proof.

1\. For any $e\in\mathcal{E}_{k}$, it follows from (4.1) and (4.2) that

$\displaystyle
e\overline{\partial}b=\sum_{i=1}^{N}\overline{\partial}(ebv^{\prime}_{i})f^{\prime}_{i}-\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})f^{\prime}_{i}b.$

Thus we have expressed $e\overline{\partial}b$ as an element of
$\Omega^{1}_{q}(G/L_{S})\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}$,
which gives us the claimed formula. The formula for the inverse map is
established analogously.

2\. For any $e\in\mathcal{E}_{-k}$, it follows from (4.1) and (4.2) that

$\displaystyle
e\overline{\partial}b=\sum_{i=1}^{N}\overline{\partial}(ebf_{i})v_{i}-\sum_{i=1}^{N}\overline{\partial}(ef_{i})v_{i}b.$

Thus we have expressed $e\overline{\partial}b$ as an element of
$\Omega^{1}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\,\mathrm{s}})$, which gives us
the claimed formula. The formula for the inverse map is established
analogously. ∎

In what follows we will denote by $\sigma_{k}$ the bimodule map associated to
the connection $\overline{\partial}_{\mathcal{E}_{k}}$ and, similarly, by
$\sigma_{-k}$ the bimodule map associated to the connection
$\overline{\partial}_{\mathcal{E}_{-k}}$.

######  Example 4.2.

Let us now check our formula for the bimodule map
$\overline{\partial}_{\mathcal{E}_{k}}$, for $k\in\mathbb{Z}_{>0}$, by looking
at the commutative case. We have

$\displaystyle\sigma_{k}\big{(}e\otimes\overline{\partial}b\big{)}$
$\displaystyle=\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i}b)\otimes
f^{\prime}_{i}-\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})\otimes
f^{\prime}_{i}b$
$\displaystyle=\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})b\otimes
f^{\prime}_{i}+\sum_{i=1}^{N}ev^{\prime}_{i}\overline{\partial}b\otimes
f^{\prime}_{i}-\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})\otimes
f^{\prime}_{i}b$ $\displaystyle=\sum_{i=1}^{N}\overline{\partial}b\otimes
ev^{\prime}_{i}f^{\prime}_{i}$ $\displaystyle=\overline{\partial}b\otimes e.$

Thus we see that $\sigma_{k}$ reduces to the usual flip map, as it should. A
similar situation holds for the line bundle $\mathcal{E}_{-k}$.

We finish with the interesting parallel observation that the generalised
determinant identities can be used to give an explicit description of the
connections $\partial_{\mathcal{E}_{k}}$ and
$\overline{\partial}_{\mathcal{E}_{k}}$. This generalises the special case of
the line module $\mathcal{E}_{1}$ over quantum projective space
$\mathcal{O}_{q}\big{(}\mathbb{CP}^{n}\big{)}$ given in [28, Example 8.5].

######  Proposition 4.3.

Let $\mathcal{O}_{q}(G/L_{S})$ be an irreducible quantum flag manifold, and
$k\in\mathbb{Z}_{>0}$, then for any $e\in\mathcal{E}_{k}$, we have

$\displaystyle\partial_{\mathcal{E}_{k}}(e)=\sum_{i=1}^{N}\partial(ev^{\prime}_{i})\otimes
f_{i}^{\prime},\qquad\overline{\partial}_{\mathcal{E}_{k}}(e)=\sum_{i=1}^{N}\overline{\partial}(ev^{\prime}_{i})\otimes
f^{\prime}_{i}.$

For any $e\in\mathcal{E}_{-k}$, we have

$\displaystyle\partial_{\mathcal{E}_{-k}}(e)=\sum_{i=1}^{N}\partial(ef_{i})\otimes
v_{i},\qquad\overline{\partial}_{\mathcal{E}_{-k}}(e)=\sum_{i=1}^{N}\overline{\partial}(ef_{i})\otimes
v_{i}.$ (4.3)

###### Proof.

We give the proof for the first identity, the proofs of the other three being
completely analogous. This follows from the calculation

$\displaystyle\partial_{\mathcal{E}_{k}}(e)=j^{-1}(\partial(e))=j^{-1}\left(\sum_{i=1}^{N}\partial(e)v^{\prime}_{i}f^{\prime}_{i}\right)=j^{-1}\left(\sum_{i=1}^{N}\partial(ev^{\prime}_{i})f^{\prime}_{i}\right)=\sum_{i=1}^{N}\partial(ev^{\prime}_{i})\otimes
f^{\prime}_{i},$

where as usual, we have used the fact that $\partial v_{i}=0$, for all
$i=1,\dots,N$. ∎

###  4.2 Some examples in terms of the FRT generators

The non-exceptional Drinfeld–Jimbo quantum coordinate algebras
$\mathcal{O}_{q}(G)$ admit a well-known alternative construction, called the
FRT construction [30], which essentially gives a generator and relation
presentation in terms of the coordinate functions of the first fundamental
representation $V_{\varpi_{1}}$ of $U_{q}(\mathfrak{g})$. It proves useful
when explicit commutation relations are required, as in Section 4.3 below, and
is commonly used in the $C^{*}$-algebra literature.

In this subsection we give an FRT presentation of the bimodule maps for the
special cases of the standard Podleś sphere, the more general quantum
Grassmannians, and the quantum quadrics, special cases of the irreducible
quantum flag manifolds, as presented in Appendix A below.

####  4.2.1 The standard Podleś sphere

The standard Podleś sphere $\mathcal{O}_{q}\big{(}S^{2}\big{)}$ is the single
quantum flag manifold of $\mathcal{O}_{q}(\mathrm{SU}_{2})$, moreover the
quantum Poisson homogeneous space of $\mathcal{O}_{q}\big{(}S^{2}\big{)}$
coincides with $\mathcal{O}_{q}(\mathrm{SU}_{2})$. Denoting by $u^{i}_{j}$,
for $i,j=1,2$, the standard generators of $\mathcal{O}_{q}(\mathrm{SU}_{2})$
(see [22, Section 4.1]), the quantum determinant relations

$\displaystyle u_{11}u_{22}-qu_{21}u_{12}=1,\qquad
u_{22}u_{11}-q^{-1}u_{12}u_{21}=1$

give a pair of relations of the form (4.1). Thus we see that for any
$e\in\mathcal{E}_{1}$, the element
$\sigma_{1}\big{(}e\otimes\overline{\partial}b\big{)}$ is equal to

$\displaystyle\overline{\partial}(ebu_{22})\otimes
u_{11}-\overline{\partial}(eu_{22})\otimes
u_{11}b-q^{-1}\big{(}\overline{\partial}(ebu_{12})\otimes
u_{21}-\overline{\partial}(eu_{12})\otimes u_{21}b\big{)}.$

Moreover, for any $e\in\mathcal{E}_{-1}$, the element
$\sigma_{-1}(e\otimes\overline{\partial}b)$ is equal to

$\displaystyle\overline{\partial}(ebu_{11})\otimes
u_{22}-\overline{\partial}(eu_{11})\otimes
u_{22}b-q\big{(}\overline{\partial}(ebu_{21})\otimes
u_{12}-\overline{\partial}(eu_{21})\otimes u_{12}b\big{)}.$

The line bundles $\mathcal{E}_{-2}$ and $\mathcal{E}_{2}$ are isomorphic as
relative line modules to the fodc $\Omega^{(1,0)}_{q}\big{(}S^{2}\big{)}$ and
$\Omega^{(0,1)}_{q}\big{(}S^{2}\big{)}$ respectively. In this case the square
of the first determinant relation

$\displaystyle(u_{22})^{2}(u_{11})^{2}-\big{(}q+q^{-1}\big{)}u_{22}u_{12}u_{21}u_{11}+q^{-2}(u_{12})^{2}(u_{21})^{2}=1,$

gives a relation of the form (4.1). This means that for $e\in\mathcal{E}_{2}$,
we have

$\displaystyle\sigma_{2}\big{(}e\otimes\overline{\partial}b\big{)}={}$
$\displaystyle\overline{\partial}\big{(}eb(u_{22})^{2}\big{)}\otimes(u_{11})^{2}-\overline{\partial}\big{(}e(u_{22})^{2}\big{)}\otimes(u_{11})^{2}b$
$\displaystyle{}-\big{(}q+q^{-1}\big{)}\big{(}\overline{\partial}(ebu_{22}u_{12}\big{)}\otimes
u_{21}u_{11}-\overline{\partial}(ebu_{22}u_{12})\otimes u_{21}u_{11}b)$
$\displaystyle{}-q^{-2}\big{(}\overline{\partial}\big{(}eb(u_{12})^{2}\big{)}\otimes(u_{21})^{2}-\overline{\partial}\big{(}e(u_{12})^{2}\big{)}\otimes(u_{21})^{2}b\big{)}.$

Moreover, we have an analogous expression for the line bundle
$\mathcal{E}_{-2}$. This reproduces the explicit presentation of the bimodule
map given by Beggs and Majid in [3, example 5.51].

####  4.2.2 The quantum grassmannians

Let us now consider the $A_{n-1}$-series irreducible quantum flag manifolds,
that is, the quantum Grassmannians, where the crossed simple root $\alpha_{m}$
corresponds to the quantum Grassmannian $\mathcal{O}_{q}(\mathrm{Gr}_{n,m})$.
Quantum projective space $\mathcal{O}_{q}\big{(}\mathbb{CP}^{n-1}\big{)}$
corresponds to the special cases of $\alpha_{m}=\alpha_{1}$, and
$\alpha_{m}=\alpha_{n-1}$. Moreover, these reduce to the standard Podleś
sphere for the case of $\mathcal{O}_{q}(\mathrm{SU}_{2})$.

Denote by $u_{ij}$, for $i,j=1,\dots,n$, the FRT generators of
$\mathcal{O}_{q}(\mathrm{SU}_{n})$ [30], (see also [22, Section 9.2]). Let
$I:=\\{i_{1},\dots,i_{m}\\}$ and $J:=\\{j_{1},\dots,j_{p}\\}$ be a pair of
subsets of $\\{1,\dots,n\\}$. The associated quantum minor $[I|J]$ is the
element of $\mathcal{O}_{q}(\mathrm{SU}_{n})$ given by

$\displaystyle[I|J]:=\sum_{\sigma\in
S_{p}}(-q)^{\ell(\sigma)}u_{\sigma(i_{1})j_{1}}\cdots
u_{\sigma(i_{p})j_{p}}=\sum_{\sigma\in
S_{p}}(-q)^{\ell(\sigma)}u_{i_{1}\sigma(j_{1})}\cdots u_{i_{p}\sigma(j_{p})}.$

The quantum Poisson homogeneous space $\mathcal{O}_{q}(S^{n,m})$ of the
quantum Grassmannian is generated by the quantum minors

$\displaystyle
z_{A}:=\big{[}A|M^{\perp}\big{]},\qquad\overline{z}_{B}:=[B|M],\qquad\textrm{for
}|A|=n-m,\ |B|=m,$

where $M:=\\{1,\dots,m\\}$ and $M^{\perp}:=\\{m+1,\dots,n\\}$.

For $I$, $J$ ordered subsets of $\\{1,\dots,n\\}$, we denote by $I^{\perp}$,
and $J^{\perp}$, the ordered complements to $I$, and $J$ respectively, in
$\\{1,\dots,n\\}$. Moreover, we write

$\displaystyle\ell(I,J):=|\\{(i,j)\in I\times
J\,|\,i>j\\}|,\qquad\ell_{\perp}(A,B):=\ell\big{(}A,A^{\perp}\big{)}-\ell\big{(}B,B^{\perp}\big{)}.$

This gives the well-known antipode expression

$\displaystyle
S([I|J])=(-q)^{\ell_{\perp}(I,J)}\big{[}{J^{\perp}}|{I^{\perp}}\big{]},$

From this we can produce the standard identity

$\displaystyle 1=\varepsilon(z_{M})$
$\displaystyle=m\circ(S\otimes\mathrm{id})\circ\Delta(z_{M})$
$\displaystyle=\sum_{|A|=m}S([M,A])[A,M]$
$\displaystyle=\sum_{|A|=m}q^{\ell_{\perp}(M,A)}\big{[}A^{\perp},M^{\perp}\big{]}[A,M]$
$\displaystyle=\sum_{|A|=m}q^{\ell_{\perp}(M,A)}\overline{z}_{A^{\perp}}z_{A},$

which is of the form (4.1). Now, for $e\in\mathcal{E}_{1}$, we see that

$\displaystyle\sigma_{1}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{|A|=n-r}q^{\ell_{\perp}(M,A)}\overline{\partial}(eb\overline{z}_{A^{\perp}})\otimes
z_{A}-\sum_{|A|=n-r}q^{\ell_{\perp}(M,A)}\overline{\partial}(e\overline{z}_{A^{\perp}})\otimes
z_{A}b.$

A similar argument shows that, for $e\in\mathcal{E}_{-1}$,

$\displaystyle\sigma_{-1}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{|B|=m}q^{\ell_{\perp}(M,B)}\overline{\partial}(ebz_{B^{\perp}})\otimes\overline{z}_{B}-\sum_{|B|=m}q^{\ell_{\perp}(M,B)}\overline{\partial}(ez_{B^{\perp}})\otimes\overline{z}_{B}b.$

Following the same approach we can also produce explicit formula for the
connections $\overline{\partial}_{\mathcal{E}_{k}}$. For for
$e\in\mathcal{E}_{1}$, it holds that

$\displaystyle\overline{\partial}_{\mathcal{E}_{-1}}(e)=\sum_{|A|=n-r}q^{\ell_{\perp}(M,A)}\overline{\partial}(e\overline{z}_{A^{\perp}})\otimes
z_{A}.$

For $e\in\mathcal{E}_{-1}$, it holds that

$\displaystyle\overline{\partial}_{\mathcal{E}_{-1}}(e)=\sum_{|B|=m}q^{\ell_{\perp}(M,B)}\overline{\partial}(ez_{B^{\perp}})\otimes\overline{z}_{B}.$

Analogous, if more technical, quantum minor type formulae for $\sigma_{k}$ and
$\partial$ can be produced for the general line bundles $\mathcal{E}_{k}$, for
$k\in\mathbb{Z}$. For the special case of the standard Podleś sphere these
formulae reduce to the presentations of the anti-holomorphic connection given
by Beggs and Majid in [3, Example 5.23].

####  4.2.3 The quantum quadrics

Let us now consider the $B_{n}$-series irreducible quantum flag manifolds,
which is to say, the _odd quantum quadrics_
$\mathcal{O}_{q}(\mathbf{Q}_{2n+1})$. In this case the Poisson quantum
homogeneous space $q$-deforms the coordinate functions of the Stieffel
manifold $V_{2}\mathbb{R}^{2n+1}\simeq\mathrm{SO}_{2n+1}/\mathrm{SO}_{2n-1}$.
Thus it is called _quantum real $(2n+1,2)$-Stieffel manifold_ and denoted by
$\mathcal{O}_{q}\big{(}V_{2}\mathbb{R}^{2n+1}\big{)}$.

For the $B$-series, the FRT algebra is a $q$-deformation of the coordinate
algebra of $\mathrm{SO}_{2n+1}$. In other words, it is the proper subalgebra
of $\mathcal{O}_{q}(\mathrm{Spin}_{2n+1})$ generated by the coordinate
functions of the first fundamental representation $V_{\varpi_{1}}$ of
$U_{q}(\mathfrak{so}_{2n+1})$. We write $u_{ij}$, where $i,j=1,\dots,2n+1$,
for the FRT generators of $\mathcal{O}_{q}(\mathrm{SO}_{2n+1})$. Denoting

$\displaystyle
i^{\prime}=2n+2-i,\qquad\rho_{i}:=n+\frac{1}{2}-i,\quad\textrm{for
}i<i^{\prime},\qquad\rho_{i}:=-\rho_{i^{\prime}},\quad\textrm{for
}i^{\prime}<i,$

an explicit formula for the antipode of $\mathcal{O}_{q}(\mathrm{SO}_{2n+1})$
is given by

$\displaystyle S(u_{ij})=q^{\rho_{i}-\rho_{j}}u_{j^{\prime}i^{\prime}},$

The quantum Stieffel manifold
$\mathcal{O}_{q}\big{(}V_{2}\mathbb{R}^{2n+1}\big{)}$ is a subalgebra of
$\mathcal{O}_{q}(\mathrm{SO}_{2n+1})$, with generators $z_{i}=u_{i,2n+1}$, and
$\overline{z}_{i}=u_{i1}$, for $i=1,\dots,2n+1$. The identity

$\displaystyle
1=\varepsilon(z_{2n+1})=m\circ(S\otimes\mathrm{id})\circ\Delta(z_{2n+1})=\sum_{a=1}^{2n+1}q^{\rho_{i}-n-\frac{1}{2}}\overline{z}_{a^{\prime}}z_{a},$

follows from the antipode formula, and implies that, for
$e\in\mathcal{E}_{1}$,

$\displaystyle\sigma_{1}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{i=1}^{2n+1}q^{-(\rho_{i}+n+\frac{1}{2})}\overline{\partial}(eb\overline{z}_{i^{\prime}})\otimes
z_{i}-\sum_{i=1}^{2n+1}q^{-(\rho_{i}+n+\frac{1}{2})}\overline{\partial}(e\overline{z}_{i})\otimes
z_{i}b.$

An analogous argument shows that, for $e\in\mathcal{E}_{-1}$, we have

$\displaystyle\sigma_{-1}\big{(}e\otimes\overline{\partial}b\big{)}=\sum_{i=1}^{2n+1}q^{\rho_{i}-n-\frac{1}{2}}\overline{\partial}(ebz_{i^{\prime}})\otimes\overline{z}_{i}-\sum_{i=1}^{2n+1}q^{\rho_{i}-n-\frac{1}{2}}\overline{\partial}(ez_{i})\otimes\overline{z}_{i}b.$

Following the same approach we can also produce explicit formula for the
connections $\overline{\partial}_{\mathcal{E}_{k}}$. For for
$e\in\mathcal{E}_{1}$, it holds that

$\displaystyle\overline{\partial}_{\mathcal{E}_{1}}(e)=\sum_{i=1}^{2n+1}q^{-(\rho_{i}+n+\frac{1}{2})}\overline{\partial}(e\overline{z}_{i^{\prime}})\otimes
z_{i}$

For $e\in\mathcal{E}_{-1}$, it holds that

$\displaystyle\overline{\partial}_{\mathcal{E}_{-1}}(e)=\sum_{i=1}^{2n+1}q^{\rho_{i}-n-\frac{1}{2}}\overline{\partial}(ez_{i^{\prime}})\otimes\overline{z}_{i}.$

The higher order bundles $\mathcal{E}_{k}$ admit analogous descriptions, as do
the _even quantum quadrics_ $\mathcal{O}_{q}(\mathbf{Q}_{2n})$, which is to
say the $C_{n}$-irreducible quantum flag manifolds.

###  4.3 A description using Takeuchi’s equivalence

Consider, for any quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, the
category of _relative Hopf modules_

${}^{~{}~{}~{}\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}_{\mathcal{O}_{q}(G/L_{S})},$

whose objects are left $\mathcal{O}_{q}(G)$-comodules
$\Delta_{L}\colon\mathcal{F}\to\mathcal{O}_{q}(G)\otimes\mathcal{F}$, endowed
with the structure of an $\mathcal{O}_{q}(G/L_{S})$-bimodule, such that

$\displaystyle\Delta_{L}(bfb^{\prime})=\Delta_{L}(b)\Delta_{L}(f)\Delta_{L}(b^{\prime}),\qquad\textrm{for
all }f\in\mathcal{F},\ b,b^{\prime}\in\mathcal{O}_{q}(G/L_{S}),$

and whose morphisms are both left $\mathcal{O}_{q}(G)$-comodule, and
$\mathcal{O}_{q}(G/L_{S})$-bimodule, maps. Moreover, we consider the full
subcategory

${}^{~{}~{}~{}\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}_{0},$

whose objects $\mathcal{F}$ are those satisfying
$\mathcal{F}\mathcal{O}_{q}(G/L_{S})^{+}=\mathcal{O}_{q}(G/L_{S})^{+}\mathcal{F}$.
This is a monoidal category with respect to the bimodule tensor product
$\otimes_{B}$. Moreover, the calculi $\Omega^{(0,1)}_{q}(G/L_{S})$ and
$\Omega^{(1,0)}_{q}(G/L_{S})$ live in this subcategory [18, Proposition 3.3],
as do the relative line modules $\mathcal{E}_{k}$, as explained for example in
[9, Appendix A.3].

Consider next the category

${}^{\mathcal{O}_{q}(L_{S})}\mathrm{Mod}$

whose objects $\Delta_{L}\colon V\to\mathcal{O}_{q}(L_{S})\otimes V$ are left
$\mathcal{O}_{q}(L_{S})$-comodules, and whose morphisms are left
$\mathcal{O}_{q}(L_{S})$-comodule maps. This has a monoidal structure given by
the usual tensor product of comodules.

We define a functor

$\Phi\colon\
{}^{~{}~{}~{}\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}_{\mathcal{O}_{q}(G/L_{S})}\to{}^{\mathcal{O}_{q}(L_{S})}\mathrm{Mod}_{\mathcal{O}_{q}(G/L_{S})},$

by setting
$\Phi(\mathcal{F}):=\mathcal{F}/\mathcal{O}_{q}(G/L_{S})^{+}\mathcal{F}$, with
the left $\mathcal{O}_{q}(L_{S})$-comodule structure of
$\Phi(\mathcal{\mathcal{F}})$ given by

$\displaystyle\Delta_{L}[f]:=\pi_{S}(f_{(-1)})\otimes[f_{(0)}],$

where the square brackets denote the coset of an element in
$\Phi(\mathcal{\mathcal{F}})$, and morphisms descend to the quotient. A
functor in the other direction

$\Psi\colon\
{}^{\mathcal{O}_{q}(L_{S})}\mathrm{Mod}_{\mathcal{O}_{q}(G/L_{S})}\to{}^{~{}~{}~{}\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}_{\mathcal{O}_{q}(G/L_{S})},$

is given by the cotensor product
$\Psi(V):=\mathcal{O}_{q}(G)\square_{\mathcal{O}_{q}(L_{S})}V$ which acts on a
morphism $\gamma$ as $\Phi(\gamma)=\mathrm{id}\otimes\gamma$.

Since $\mathcal{O}_{q}(G)$ is faithfully flat as a right
$\mathcal{O}_{q}(G/L_{S})$-module, it follows from Takeuchi’s equivalence for
quantum homogeneous spaces [32], that $\Phi$ induces an equivalence of
categories. Moreover, it is a monoidal equivalence [29, Section 4]. We note
that the relative line modules $\mathcal{E}_{k}$ are invertible objects in the
category
${}^{~{}~{}~{}\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}_{0}$.
Indeed, as shown in [9, Proposition 5.7] for details), they are the only such
objects.

Since any $\sigma_{k}$ is clearly a morphism, we can completely describe it by
identifying its image under $\Phi$. The following proposition tells us that
$\Phi(\sigma_{k})$ takes quite a simple form.

For sake of clarity in the proof, we find it useful to establish a simple
general lemma.

######  Lemma 4.4.

Let $\nabla\colon\mathcal{F}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}$ and
$\nabla^{\prime}\colon\mathcal{F}^{\prime}\to\Omega^{1}(B)\otimes_{B}\mathcal{F}^{\prime}$
be two bimodule connections, and
$\alpha\colon\mathcal{F}\to\mathcal{F}^{\prime}$ an isomorphism between them.
We note that the following diagram commutes

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
27.79713pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{F}\otimes_{B}\Omega^{1}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{\alpha\otimes\mathrm{id}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\sigma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.95602pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
51.79713pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 113.95602pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{1}(B)\otimes_{B}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
140.42763pt\raise-20.56888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{\mathrm{id}\otimes\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 140.42763pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{F}^{\prime}\otimes_{B}\Omega^{1}(B)\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
64.66386pt\raise-46.20026pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{\sigma^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
111.79713pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
51.79713pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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81.79713pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
111.79713pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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(4.4)

where $\sigma$ and $\sigma^{\prime}$ are the bimodule maps associated to
$\nabla$ and $\nabla^{\prime}$ respectively.

###### Proof.

The calculation

$\displaystyle\sigma^{\prime}(\alpha(f)\otimes\mathrm{d}b)=\nabla^{\prime}(\alpha(fb))-\nabla^{\prime}(\alpha(f))b=(\mathrm{id}\otimes\alpha)(\nabla(fb)-\nabla(f)b)=(\mathrm{id}\otimes\alpha)\sigma(f\otimes\mathrm{d}b)$

directly implies the result. ∎

######  Proposition 4.5.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, the map

$\Phi(\sigma_{k})\colon\
\Phi(\mathcal{E}_{k})\otimes\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\to\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\otimes\Phi(\mathcal{E}_{k}),$

for any line module $\mathcal{E}_{k}$, with $k\in\mathbb{Z}$, satisfies

$\Phi(\sigma_{k})([e]\otimes[\omega])=\theta^{k}[\omega]\otimes[e],$

for a nonzero scalar $\theta\in\mathbb{C}$.

###### Proof.

Since the functor $\Phi$ induces a monoidal equivalence, we have an
isomorphism

$\displaystyle\Phi\big{(}\mathcal{E}_{k}\otimes_{\mathcal{O}_{q}(G/L_{S})}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\simeq\Phi(\mathcal{E}_{k})\otimes\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}.$

As in the classical setting, for each irreducible quantum flag manifold,
$\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}$ is an irreducible
$U_{q}(\mathfrak{l}_{S})$-module. Moreover, since $\mathcal{E}_{k}$ is
invertible as a $B$-bimodule, the $U_{q}(\mathfrak{l}_{S})$-module
$\Phi(\mathcal{E}_{k})$ must be one-dimensional. Thus we see that
$\Phi(\mathcal{E}_{k})\otimes\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}$ is
an irreducible $U_{q}(\mathfrak{l}_{S})$-module. An analogous argument
establishes that the tensor product
$\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\otimes\Phi(\mathcal{E}_{k})$ is
irreducible as a $U_{q}(\mathfrak{l}_{S})$-module. If
$\big{[}\,\overline{\partial}b_{hw}\big{]}$ is a highest weight vector of
$\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}$, then there exists a non-zero
scalar $\theta_{k}$, such that

$\Phi(\sigma_{k})\big{(}\big{[}\,\overline{\partial}b_{hw}\big{]}\otimes[e]\big{)}=\theta_{k}[e]\otimes\big{[}\,\overline{\partial}b_{hw}\big{]}.$

For any other element $\big{[}\,\overline{\partial}b\big{]}$, there exists an
$X\in U_{q}\big{(}\mathfrak{l}_{S}^{\,\mathrm{s}}\big{)}$ such that

$\displaystyle\big{[}\,\overline{\partial}b\big{]}\otimes[e]=X\big{(}\big{[}\,\overline{\partial}b_{hw}\big{]}\otimes[e]\big{)}.$

Hence we have that

$\displaystyle\Phi(\sigma_{k})\big{(}\big{[}\,\overline{\partial}b\big{]}\otimes[e]\big{)}$
$\displaystyle=\theta_{k}X\big{(}[e]\otimes\big{[}\,\overline{\partial}b_{hw}\big{]}\big{)}=\theta_{k}(X_{(1)}[e])\otimes\big{(}X_{(2)}\big{[}\,\overline{\partial}b_{hw}\big{]}\big{)}$
$\displaystyle=\theta_{k}[e]\otimes\big{(}X\big{[}\,\overline{\partial}b_{hw}\big{]}\big{)}=\theta_{k}[e]\otimes\big{[}\,\overline{\partial}b\big{]}.$

Note that the penultimate equality follows from the fact that, since
$\mathfrak{l}_{S}^{\,\mathrm{s}}$ is a semisimple Lie algebra, it admits no
non-trivial one-dimensional representations, and so, the Hopf algebra
$U_{q}\big{(}\mathfrak{l}_{S}^{\,\mathrm{s}}\big{)}$ admits no non-trivial
$1$-dimensional representations.

Now since the connection $\overline{\partial}_{\mathcal{E}_{k}}$ is isomorphic
to the $k$-fold tensor product of the connection
$\overline{\partial}_{\mathcal{E}_{1}}$, it follows from (4.4) that we have
the commutative diagram

$\textstyle{\Phi(\mathcal{E}_{k})\otimes\Phi(\mathcal{E}_{l})\otimes\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(\sigma_{k}\otimes\mathrm{id})\,\circ\,\Phi(\mathrm{id}\otimes\sigma_{l})}$$\scriptstyle{\simeq\,\otimes\mathrm{id}}$$\textstyle{\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\otimes\Phi(\mathcal{E}_{k})\otimes\Phi(\mathcal{E}_{l})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\otimes\,\simeq}$$\textstyle{\Phi(\mathcal{E}_{k+l})\otimes\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(\sigma_{k+l})}$$\textstyle{\Phi\big{(}\Omega^{(0,1)}_{q}(G/L_{S})\big{)}\otimes\Phi(\mathcal{E}_{k+l}).}$

We now see, for $e\in\mathcal{E}_{k+l}$, $b\in\mathcal{O}_{q}(G/L_{S})$, that

$\displaystyle\Phi(\sigma_{k+l})\big{(}[e]\otimes\big{[}\,\overline{\partial}b\big{]}\big{)}=\theta_{k}\theta_{l}\big{[}\,\overline{\partial}b\big{]}\otimes[e].$
(4.5)

Recalling next Example 2.1, we see that

$\displaystyle\Phi(\sigma_{1})\big{(}[1]\otimes\big{[}\,\overline{\partial}b\big{]}\big{)}=\big{[}\,\overline{\partial}b\big{]}\otimes[1].$

It now follows from 4.5 that $\theta_{-1}=\theta_{1}^{-1}$, which in turn
implies that the claimed nonzero scalar $\theta\in\mathbb{C}$ is given by
$\theta_{1}$. ∎

######  Example 4.6.

In this example we determine the constant $\theta$ for the standard Podleś
sphere $\mathcal{O}_{q}\big{(}S^{2}\big{)}$. It follows from (4.3) that, for
$e\in\mathcal{E}_{1}$, the element
$\Phi(\sigma_{1})\big{[}e\otimes\overline{\partial}b\big{]}$ is equal to

$\displaystyle\Phi(\sigma_{1})\big{[}e\otimes\overline{\partial}b\big{]}={}$
$\displaystyle\big{[}\,\overline{\partial}(ebu_{11})\otimes
u_{22}\big{]}-\big{[}\,\overline{\partial}(eu_{11})\otimes u_{22}b\big{]}$
$\displaystyle{}-q\big{(}\big{[}\,\overline{\partial}(ebu_{21})\otimes
u_{12}\big{]}-q\big{[}\,\overline{\partial}(eu_{21})\otimes
u_{12}b\big{]}\big{)}.$

Choosing $e=u_{22}$ and $b=u_{12}u_{22}$, we get

$\displaystyle\Phi(\sigma_{1})\big{[}u_{22}\otimes\overline{\partial}(u_{12}u_{22})\big{]}$
$\displaystyle=\big{[}\,\overline{\partial}(u_{22}u_{12}u_{22}u_{11})\otimes
u_{22}\big{]}$
$\displaystyle=\big{[}\,\overline{\partial}(u_{22}u_{12})u_{22}u_{11}\otimes
u_{22}\big{]}+\big{[}u_{22}u_{12}\overline{\partial}(u_{22}u_{11})\otimes
u_{22}\big{]}$
$\displaystyle=q^{-1}\big{[}\,\overline{\partial}(u_{12}u_{22})\otimes
u_{22}\big{]}.$

From which it follows that $\theta=q^{-1}$, showing in particular that
$\theta$ is not equal to $1$. This calculation can be extended to the quantum
Grassmannians, and the quantum quadrics using the FRT presentation of the
bimodule map $\sigma$ given in Section 4.2.

######  Proposition 4.7.

It holds that

$\sigma_{k}\big{(}e\otimes\overline{\partial}b\big{)}=\theta^{k}e_{(-2)}b_{(1)}S\big{(}e_{(-1)}\big{)}S\big{(}b_{(2)}\big{)}\overline{\partial}b_{(3)}\otimes
e_{(0)},$

for any $b\in\mathcal{O}_{q}(G/L_{S})$, and $e\in\mathcal{E}_{k}$.

###### Proof.

A unit for the equivalence is given by

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

for any relative Hopf module $\mathcal{F}$. Thus we have that

$\displaystyle\sigma_{k}(e\otimes\overline{\partial}b)$
$\displaystyle=\mathrm{U}^{-1}\circ(\mathrm{id}\otimes\Phi(\sigma_{k}))\circ\mathrm{U}\big{(}e\otimes\overline{\partial}b\big{)}$
$\displaystyle=\mathrm{U}^{-1}\circ(\mathrm{id}\otimes\Phi(\sigma_{k}))\big{(}e_{(-1)}b_{(1)}\otimes\big{[}e_{(0)}\otimes\overline{\partial}b_{(2)}\big{]}\big{)}$
$\displaystyle=\theta^{k}\mathrm{U}^{-1}\big{(}e_{(-1)}b_{(1)}\otimes\big{[}\,\overline{\partial}b_{(2)}\otimes
e_{(0)}\big{]}\big{)}.$

Recalling the explicit presentation of the inverse of $\mathrm{U}$ given in
[29, Corollary 2.7], we see that

$\displaystyle\theta^{k}\mathrm{U}^{-1}\big{(}e_{(-1)}b_{(1)}\otimes\big{[}\,\overline{\partial}b_{(2)}\otimes
e_{(0)}\big{]}\big{)}$
$\displaystyle=\theta^{k}e_{(-2)}b_{(1)}S\big{(}b_{(2)}e_{(-1)}\big{)}\overline{\partial}b_{(3)}\otimes
e_{(0)}$
$\displaystyle=\theta^{k}e_{(-2)}b_{(1)}S\big{(}e_{(-1)}\big{)}S\big{(}b_{(2)}\big{)}\overline{\partial}b_{(3)}\otimes
e_{(0)},$

giving us the claimed identity. ∎

It is interesting to note that in the commutative case, it follows from the
antipode axiom for a Hopf algebra that the formula in Proposition 4.7 reduces
to the flip map, as it should.

##  Appendix A Table of simple roots for the irreducible quantum
flag manifolds

In this appendix we recall, for the reader’s convenience, an explicit
description of the irreducible quantum flag manifolds. In the classical
setting the irreducible flag manifolds are those flag manifolds $G/L_{S}$ for
which the space of anti-holomorphic forms $\Omega^{(0,1)}_{q}(G/L_{S})$ is
irreducible as a $G$-module. The corresponding subset of simple roots is
always of the form $\Pi\backslash\\{\alpha_{x}\\}$, such that $\alpha_{x}$ has
coefficient $1$ in the highest root of $\mathfrak{g}$. In Table 1 $\alpha_{x}$
is denoted graphically by a coloured node in the Dynkin diagram of
$\mathfrak{g}$, where the nodes are numbered according to [19, Section 11.4].

Table 1:  Series | $\mathcal{O}_{q}(G)$ | Crossed node | $\mathcal{O}_{q}(G/L_{S})$
---|---|---|---
$A_{n}$ | $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ |  | $\mathcal{O}_{q}(\mathrm{Gr}_{n+1,m})$
$B_{n}$ | $\mathcal{O}_{q}(\mathrm{Spin}_{2n+1})$ |  | $\mathcal{O}_{q}(\mathbf{Q}_{2n+1})$
$C_{n}$ | $\mathcal{O}_{q}(\mathrm{Sp}_{n})$ |  | $\mathcal{O}_{q}(\mathbf{L}_{n})$
$D_{n}$ | $\mathcal{O}_{q}(\mathrm{Spin}_{2n})$ |  | $\mathcal{O}_{q}(\mathbf{Q}_{2n})$
$D_{n}$ | $\mathcal{O}_{q}(\mathrm{Spin}_{2n})$ |  | $\mathcal{O}_{q}(\textbf{S}_{n})$
$E_{6}$ | $\mathcal{O}_{q}(E_{6})$ |  | $\mathcal{O}_{q}(\mathbb{OP}^{2})$
$E_{7}$ | $\mathcal{O}_{q}(E_{7})$ |  | $\mathcal{O}_{q}(\textbf{F})$

#### Acknowledgements

AC was supported by the GAČR project 20-17488Y and RVO: 67985840. AC and RÓB
are supported by the Charles University PRIMUS grant _Spectral Noncommutative
Geometry of Quantum Flag Manifolds_ PRIMUS/21/SCI/026. We would like to thank
Henrik Winther for useful discussions. Moreover, we would like to thank the
referees for their careful readings of the paper and their helpful
suggestions.

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