# Principal Pairs of Quantum Homogeneous Spaces

Alessandro Carotenuto  Institute of Mathematics, Czech Academy of Sciences,
Žitná 25, 115 67 Prague, Czech Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Réamonn Ó Buachalla  Mathematical Institute of
Charles University, Sokolovská 83, Prague, Czech Republic
<EMAIL_ADDRESS>
###### Abstract.

We propose a simple but effective framework for producing examples of
covariant faithfully flat (generalised) Hopf–Galois extensions from a nested
pair of quantum homogeneous spaces. Our construction is modelled on the
classical situation of a homogeneous fibration $G/N\to G/M$, for $G$ a group,
and $N\subseteq M\subseteq G$ subgroups. Variations on Takeuchi’s equivalence
and Schneider’s descent theorem are presented in this context. Quantum flag
manifolds and their associated quantum Poisson homogeneous spaces are taken as
motivating examples. Moreover, a large collection of _noncommutative
fibrations_ (in the spirit of Brzeziński and Szymański) are constructed.

AC is supported by the GAČR project 20-17488Y and RVO: 67985840. RÓB was
supported by GAČR through the framework of the grant GA19-06357S, and is
supported by the Charles University PRIMUS grant _Spectral Noncommutative
Geometry of Quantum Flag Manifolds_ PRIMUS/21/SCI/026

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of the Paper
  2. 2 Preliminaries on Principal Comodule Algebras
    1. 2.1 Principal Comodule Algebras
    2. 2.2 Schneider’s Descent Theorem
    3. 2.3 Relative Hopf Modules
    4. 2.4 Takeuchi’s Equivalence
  3. 3 Principal Pairs of Quantum Homogeneous Spaces
    1. 3.1 Nested Pairs of Quantum Homogeneous Spaces
    2. 3.2 A Generalisation of Takeuchi’s Equivalence for Nested Pairs
    3. 3.3 Some Consequences of the Equivalence
    4. 3.4 Principal Pairs
    5. 3.5 Schneider’s Equivalence for Principal Pairs
  4. 4 Group Algebra Principal Pairs and Relative Line Modules
    1. 4.1 Group Algebra Principal Pairs
    2. 4.2 Relative Line Modules and Group Algebra Principal Pairs
    3. 4.3 Schneider’s Equivalence for Group Algebra Principal Pairs
  5. 5 Quantum Flag Manifolds
    1. 5.1 Drinfeld–Jimbo Quantum Groups
    2. 5.2 Type-$1$ Representations
    3. 5.3 Quantum Levi Subalgebras
    4. 5.4 Quantum Coordinate Algebras and the Quantum Flag Manifolds
    5. 5.5 The Associated Quantum Poisson Homogeneous Spaces
    6. 5.6 Quantum Flag Manifolds and Principal Pairs
    7. 5.7 Relative Line Modules
    8. 5.8 A Principal $\ell$-Map
    9. 5.9 Non-Cleftness of the Hopf–Galois Extension
    10. 5.10 Schneider’s Equivalence and a Family of Examples
  6. 6 Examples of Non-Principal Noncommutative Fibrations
    1. 6.1 Non-Principal Examples with Irreducible Quantum Flag Manifold Bases
    2. 6.2 Noncommutative Sphere Fibrations and Quantum Stiefel Manifolds
  7. A Some Comments on Quantum Homogeneous Spaces
    1. A.1 Quantum Homogeneous Spaces as Principal Comodule Algebras
    2. A.2 Quantum Homogeneous Spaces, Ideals, and Quantum Subgroups
    3. A.3 Simple Objects of the Category ${}^{A}_{B}\mathrm{Mod}_{0}$
    4. A.4 Connections and Projectivity

##  1\. Introduction

From their purely algebraic origins [14, 25], Hopf–Galois extensions have
become a central structure in noncommutative geometry where they are rightly
considered as noncommutative analogues of principal bundles [3]. More
recently, efforts have begun to extend this work to a theory of noncommutative
fiber bundles. This has focused on the study of noncommutative sphere bundles
from both a Hopf algebraic [11] and a $C^{*}$-algebraic point of view [2]. In
this paper we propose a simple but effective new framework for producing
examples of noncommutative fibrations, both principal and non-principal, from
a nested pair of quantum homogeneous spaces. Our construction generalises the
prototypical situation of a homogeneous fibration, probably the simplest type
of fibration after a homogeneous space: let $G$ be a group, then we have a
fibration

$\displaystyle G/N\twoheadrightarrow G/M,$ $\displaystyle\textrm{ for any two
subgroups }N\subseteq M\subseteq G.$

This fibration is principal if and only if $N$ is a normal subgroup of $M$,
see for example [19]. We observe that this construction extends directly to
the Hopf algebra setting, providing a large and natural family of principal
comodule algebras. In more detail, we start with a nested pair of quantum
homogeneous spaces $B\subseteq P$ (coideal subalgebras satisfying a faithful
flatness condition) and show that $P$ is a principal $\pi_{B}(P)$-comodule
algebra with base $B$, where $\pi_{B}(P)$ is a Hopf algebra of coinvariants
playing the role of the quotient group $M/N$. Moreover, we have direct
generalisations of Takeuchi’s and Schneider’s categorical equivalences to this
setting. We call such a pair of quantum homogeneous spaces $(B,P)$ a
_principal pair_.

The space of coinvariants $\pi_{B}(P)$ is of course not guaranteed to have a
coalgebra structure, just as $N$ is not guaranteed to be normal in $M$.
However, even then the pair $B\subseteq P$ retains many attractive algebraic
properties: $P$ is a faithfully flat extension of $B$ and the extension
satisfies a generalised Hopf–Galois condition. The generalised version of
Takeuchi’s equivalence still holds and $P$ can be described as an
$(A,B)$-relative module associated to the fiber $\pi_{B}(P)$. This list of
properties places these examples firmly within the domain of Brzeziński and
Szymański’s putative theory of _noncommutative fiber bundles_ , providing a
rich family of examples deserving to be considered as _noncommutative
fibrations_.

Of particular concern to this paper is the construction of principle comodule
$\mathcal{O}(\mathbb{T}^{k})$-algebras, where $\mathbb{T}^{k}$ is the
$k$-torus. We are motivated by the classical situation of a principal torus
fibration $G/L^{\mathrm{s}}\to G/L$, where $G$ is a compact Lie group, $L$ is
a reductive subgroup, and $L^{\mathrm{s}}$ is the semisimple part of $L$. We
start with a quantum homogeneous space $B=A^{\mathrm{co}(H)}$, where we assume
that $H$ is cosemisimple and all of its grouplike elements are central. We
then consider $K_{H}$ the Hopf algebra quotient of $H$ by the augmentation
ideal of $\Gamma_{H}$, the group Hopf algebra of the group of grouplike
elements of $H$. The pair of quantum homogeneous spaces $B$ and $P$ associated
to the projections $\pi_{H}:A\to H$ and $\pi_{A}:A\to K_{H}$ is then shown to
be a principal pair, which we call a _group algebra principal pair_. We also
give a formal treatment of relative line modules in terms of an algebra
grading over the group of grouplike elements of $H$.

In the Hopf algebraic approach to noncommutative geometry, Majid and
Brzeziński’s quantum principal bundles and principal connections play a
central role, see for example the recent monograph [3]. An alternative
definition of a principal pair was introduced by Mrozinski and the authors in
[13] as a formal framework in which to construct quantum principal bundles and
quantum principal connections over the quantum Grassmannians. The fact that
any nested pair of quantum homogeneous spaces gives a principal comodule
algebra easily implies the equivalence of the two definitions. This gives
support to the framework introduced in [13] while offering a significant
simplification of the setup, providing greater clarify, and elucidating the
relationship with the classical situation.

One of our main motivations for introducing group algebra principal pairs is
to provide a firmly Hopf algebraic framework for studying relative line
modules and principal connections over quantum flag manifolds
$\mathcal{O}_{q}(G/L_{S})$. Taking the quantum Poisson homogeneous space
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ as the total space, the homogeneous
spaces $\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ and $\mathcal{O}_{q}(G/L_{S})$
are easily shown to give a group algebra principal pair. This connects the
quantum flag manifolds to the powerful machinery of principal comodule
algebras and quantum principal bundles. Concretely, we get a graded algebra
structure on the quantum Poisson homogeneous space whose homogeneous summands
are relative line modules, and every relative line module is of this form.
Moreover, we get a very useful description of the line modules in terms of the
standard generators of $\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$ (as first
considered by Stokman [39]).

The principal pair presentation of the quantum flag manifolds leads to a
number of important applications in accompanying papers. The first is the
description of the Heckenberger–Kolb calculi [21, 22] in quantum principal
bundle terms [12], generalising the case of quantum projective space given in
[32], and the more general quantum Grassmannian case given in [13]. As for the
quantum Grassmannian case, this provides a framework in which the Borel–Weil
theorem for the irreducible flag manifolds can be $q$-deformed [12]. The
Borel–Weil theorem has itself a number of important consequences. It allows us
to identify which line modules over the irreducible quantum flag manifolds are
positive and which are negative [16]. In turn, in [34] the negative line
modules are used to twist the Dolbeault–Dirac operators of the irreducible
quantum flag manifolds so as to produce Fredholm operators. In forthcoming
work, the principal comodule algebra presentation of
$\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$ will be used to present the
$C^{*}$-completion of $\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$ as a
Cuntz–Pimsner algebra, generalising the case of quantum projective space
treated in [1].

###  1.1. Summary of the Paper

The paper is organised as follows. In Section 2 we recall the standard
definitions and results about principal comodule algebras, Schneider’s descent
theorem, and Takeuchi’s equivalence for relative Hopf modules.

In Section 3, we prove a faithful flatness condition, as well as a generalised
Hopf–Galois condition, for a nested pair of quantum homogeneous spaces.
Building on this, the notion of a principal is then introduced. Principal
pairs are shown to give principal comodule algebras and hence we regain the
earlier formulation of a principal pair introduced by the authors and
Mrozinski. Along the way we establish a generalisation Takeuchi’s equivalence
for nested pairs of quantum homogeneous spaces, as well as a refinement of
Schneider’s equivalence taking into account the extra symmetry given by a
principal pair.

In Section 4 we introduce the definition of a group algebra principal pair.
More explicitly, we prove the following theorem.

######  Theorem 1.1.

Let $A$ and $H$ be Hopf algebras, $\pi_{H}:A\to H$ a surjective Hopf algebra
map, and $\mathrm{proj}_{K}:H\to K_{H}$ the canonical projection. If $A$ is
cosemisimple and all the grouplike elements of $H$ are central in $H$, then
the pair

$\left(\pi_{H}:A\to H,\,\mathrm{proj}_{K}\circ\pi_{H}:A\to K_{H}\right)$

is a principal pair.

We then show that we have an associated decomposition of the total space
$P=A^{\mathrm{co}(K_{H})}$ into a graded algebra whose summands are relative
line modules, and show that every relative line module over the base space is
of this form.

In Section 5 we treat our motivating family of examples, the quantum flag
manifolds. Beginning with the necessary definitions of Drinfeld–Jimbo
quantised enveloping algebras, quantum coordinate algebras, and the quantum
flag manifolds, we then verify the group algebra principal pair condition by
establishing the following theorem.

######  Theorem 1.2.

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

$\Big{(}\mathcal{O}_{q}(G/L_{S}),\,\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\Big{)}$

is a group algebra principal pair.

This gives a concise description of the relative line modules in terms of a
$\mathcal{P}_{S^{c}}$-algebra grading on
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$. We then use this description to
construct an explicit left $A$-covariant principal connection for the
principal comodule algebra, generalising the well-known $q$-monopole
connection for the special case of the Podleś sphere. Non-cleftness of the
Hopf–Galois extension is also established.

In Section 6, we produce two families of _noncommutative fibrations_ in the
spirit of Brzeziński and Szymański, each associated to a nested pair of
subsets of the simple roots of a complex semisimple Lie algebra
$\mathfrak{g}$. Examples include noncommutative fibrations with quantum
sphere, quantum Stiefel manifold, and quantum flag manifold fibers.

We finish with an appendix discussing some relevant details of quantum
homogeneous spaces.

### Acknowledgments

We would like to thank Andrey Krutov, Karen Strung, Hans–Jurgen Schneider,
Paolo Saracco, Rita Fioresi, Marco Matassa, Tomasz Brzeziński, and Wojciech
Szymański for helpful discussions.

##  2\. Preliminaries on Principal Comodule Algebras

In this section we the necessary results on principal comodule algebras,
Schneider’s equivalence, relative Hopf modules, and Takeuchi’s equivalence.
All this material is by now quite well known, and a more detailed presentation
can be found in the monographs [3, 7].

Throughout this section, and indeed the paper, all algebras will be unital and
defined over $\mathbb{C}$, and all Hopf algebras will be assumed to have a
bijective antipode. We denote the coproduct, counit, and antipode of a Hopf
algebra $H$ by $\Delta,\epsilon$, and $S$ respectively, and we denote the
cotensor product over $H$ by $\square_{H}$.

###  2.1. Principal Comodule Algebras

A right $H$-comodule algebra $(P,\Delta_{R})$ is said to be a $H$-Hopf–Galois
extension of $B:=P^{\mathrm{co}(H)}$ if for $m_{P}:P\otimes_{B}P\to P$ the
multiplication of $P$, a bijection is given by

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

We call $P$ the _total algebra_ and $B$ the _base algebra_. We say that $P$ is
faithfully flat as a right $B$-module if the functor

$P\otimes_{B}-:{}_{B}\mathrm{Mod}\to\mathrm{Mod_{\mathbb{C}}}$

preserves and reflects exact sequences. Faithful flatness as a left $B$-module
is defined analogously.

######  Definition 2.1.

A principal right $H$-comodule algebra is a right $H$-comodule algebra
$(P,\Delta_{R})$ such that $P$ is a Hopf–Galois extension of
$B:=P^{\mathrm{co}(H)}$ and $P$ is faithfully flat as a right and left
$B$-module.

As shown in [4, 6], a right $H$-comodule algebra $(P,\Delta_{R})$ is principal
if and only if there exists a principal $\ell$-map $\ell:H\to P\otimes P$, as
defined below.

######  Definition 2.2.

For a right $H$-comodule algebra $(P,\Delta_{R})$, a principal $\ell$-map is a
linear map $\ell:H\to P\otimes P$ satisfying

  1. 1.

$\ell(1_{H})=1_{P}\otimes 1_{P}$,

  2. 2.

$m_{P}\circ\ell=\varepsilon_{H}1_{P}$,

  3. 3.

$(\ell\otimes\mathrm{id}_{H})\circ\Delta=(\mathrm{id}_{P}\otimes\Delta_{R})\circ\ell$,

  4. 4.

$(\mathrm{id}_{H}\otimes\ell)\circ\Delta=(\Delta_{L}\otimes\mathrm{id}_{P})\circ\ell$,

where, for $\text{flip}:P\otimes H\to H\otimes P$ the flip map,
$\Delta_{L}:=(S\otimes\mathrm{id})\circ\text{flip}\circ\Delta_{R}$.

###  2.2. Schneider’s Descent Theorem

Let $H$ be a Hopf algebra, $(P,\Delta_{R})$ a right $H$-comodule algebra, and
denote $B:=P^{\mathrm{co}(H)}$. Then $(P,\Delta_{R})$ is a principal comodule
algebra if and only if the functor

$P\otimes_{B}-:{}_{B}\mathrm{Mod}\to{}_{P}\mathrm{Mod}^{H}$

induces an equivalence of categories. In this case the inverse functor to
$P\otimes_{B}-$ is

$\displaystyle{}_{P}\mathrm{Mod}^{H}\to{}_{B}\mathrm{Mod},$
$\displaystyle\mathcal{F}\mapsto\mathcal{F}^{\mathrm{co}(H)}.$

This result is known as _Schneider’s descent theorem_ , but we also find it
convenient to refer to it as _Schneider’s equivalence_ , see [38, Theorem I]
for a proof. Explicitly, the unit natural isomorphism for the equivalence is
given by

(1)
$\displaystyle\mathrm{U}:\mathcal{M}\to(P\otimes_{B}\mathcal{M})^{\mathrm{co}(\pi_{B}(P))},$
$\displaystyle m\mapsto 1\otimes m,$

while the counit isomorphism is given by

(2)
$\displaystyle\qquad\mathrm{C}:P\otimes_{B}\mathcal{N}^{\mathrm{co}(\pi_{B}(P))}\to\mathcal{N},\,$
$\displaystyle p\otimes n\mapsto pn.$

###  2.3. Relative Hopf Modules

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
$B^{+}A=AB^{+}$. (See Appendix A.2 for some equivalent presentations of the
definition.) It follows from [41, Theorem 1] that, for the Hopf algebra
surjection $\pi_{B}:A\to A/B^{+}A$, and the associated right
$\pi_{B}(A)$-coaction
$\Delta_{R,\pi_{B}}:=(\mathrm{id}\otimes\pi_{B})\circ\Delta$, the space of
coinvariants $A^{\mathrm{co}(A/B^{+}A)}$ is equal to $B$.

We denote by ${}^{A}_{B}\mathrm{Mod}$ the category of _relative Hopf modules_
, that is, the category whose objects are left $A$-comodules
$\Delta_{L}:\mathcal{F}\to A\otimes\mathcal{F}$, endowed with a left
$B$-module structure such that, for all $f\in\mathcal{F},$ $b\in B$, we have
$\Delta_{L}(bf)=\Delta_{L}(b)\Delta_{L}(f)$, and whose morphisms are left
$A$-comodule, left $B$-module, maps. As explained in Appendix A.4, very
relative Hopf module is projective as a left $B$-module. A _relative line
module_ over $B$ is an invertible object ${\mathcal{E}}$ in the cateogry
${}^{A}_{B}\mathrm{Mod}_{0}$.

###  2.4. Takeuchi’s Equivalence

In the following we denote by ${}^{\pi_{B}}\mathrm{Mod}$ the category whose
objects are left $\pi_{B}(A)$-comodules, and whose morphisms are left
$\pi_{B}(A)$-comodule maps. We use similar notation for right
$\pi_{B}(A)$-comodules.

Define a functor $\Phi:{}^{A}_{B}\mathrm{Mod}\to{}^{\pi_{B}}\mathrm{Mod}$ by
setting $\Phi(\mathcal{F}):=\mathcal{F}/B^{+}\mathcal{F}$, for
$B^{+}:=B\cap\ker(\varepsilon)$, 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 use the cotensor product $\square_{\pi_{B}(A)}$, which we
find convenient to denote by $\square_{\pi_{B}}$. Define a functor
$\Psi:{}^{\pi_{B}}\mathrm{Mod}\to{}^{A}_{B}\mathrm{Mod}$ by setting
$\Psi(V):=A\,\square_{\pi_{B}}V$, where the left $B$-module and left
$A$-comodule structures of $\Psi(V)$ are defined on the first tensor factor,
and if $\gamma$ is a morphism in ${}^{\pi_{B}}\mathrm{Mod}$, then
$\Psi(\gamma):=\mathrm{id}\otimes\gamma$. Note that $A$ is naturally an object
in ${}^{A}_{B}\mathrm{Mod}$, and that $\Phi(A)=\pi_{P}(A).$

As established in [41, Theorem 1], an adjoint equivalence of categories
between ${}^{A}_{B}\mathrm{Mod}$ and ${}^{\pi_{B}}\mathrm{Mod}$, which we call
_Takeuchi’s equivalence_ , is given by the functors $\Phi$ and $\Psi$, the
unit natural isomorphism
$\mathrm{U}:{\mathcal{F}}\to\Psi\circ\Phi({\mathcal{F}})$, defined by
$\mathrm{U}(f)=f_{(-1)}\otimes[f_{(0)}],$ and the counit natural isomorphism
$\mathrm{C}:=\varepsilon\otimes\mathrm{id}:\Phi\circ\Psi(V)\to V.$ The
_dimension_ $\mathrm{dim}({\mathcal{F}})$ of an object
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}$ is the vector space dimension of
$\Phi({\mathcal{F}})$.

Consider now the category ${}^{A}_{B}\textrm{Mod}_{0}$ whose objects are
objects ${\mathcal{F}}$ in ${}^{A}_{B}\textrm{Mod}$ endowed with the right
$B$-module structure

$\displaystyle fb:=f_{(-2)}bS(f_{(-1)})f_{(0)},$ $\displaystyle\textrm{ for
}f\in{\mathcal{F}},\,b\in B.$

It is instructive to note that any $b\in B$ acts on
$A\square_{\pi_{B}}\Phi({\mathcal{F}})$ as left multiplication on the first
tensor factor. Clearly, ${}^{A}_{B}\textrm{Mod}_{0}$ is equivalent to
${}^{A}_{B}\textrm{Mod}$, however ${}^{A}_{B}\textrm{Mod}_{0}$ comes equipped
with a monoidal structure given by the tensor product $\otimes_{B}$. Moreover,
with respect to the obvious monoidal structure on ${}^{H}\mathrm{Mod}$,
Takeuchi’s equivalence is easily endowed with the structure of a monoidal
equivalence (see [33, §4]). An immediate implication is that an object
${\mathcal{E}}$ is invertible (that is, it is a relative line module) if and
only if $\dim({\mathcal{E}})=1$.

##  3\. Principal Pairs of Quantum Homogeneous Spaces

In this section we consider nested pairs of quantum homogeneous spaces
$B\subseteq P$, which we consider as noncommutative generalisations of
homogeneous fibrations (as discussed in the introduction). We show that $P$
can be presented as a relative Hopf module associated to a homogeneous fiber,
and that $P$ is a type of generalised Hopf–Galois extension of $B$. Thus these
pairs satisfy some of the key properties Brzeziński and Szymański identified
for _noncommutative fibrations_ [11]. We then consider the case where the
fiber is a Hopf algebra, generalising homogeneous principle bundles as
considered in classical geometry [19, §2]. In this case $P$ is shown to be a
principal comodule algebra with base $B$, and hence we recover the definition
of a principal pair introduced by Mrozinski and the authors in [13]. The new
formulation offers a significant simplification of the original definition,
and sets it in the context of classical homogeneous fibrations. We also show
how Schneider’s equivalence for principal comodule algebras interacts with the
generalisation of Takeuchi’s equivalence, and give necessary and sufficient
criteria for our Hopf–Galois extensions to be non-cleft.

###  3.1. Nested Pairs of Quantum Homogeneous Spaces

Let $A$ be a Hopf algebra, and $B\subseteq P\subseteq A$ a nested pair of
quantum homogeneous $A$-spaces. Note that $P$ is naturally an object in
${}^{A}_{B}\mathrm{Mod}$ so that the unit of Takeuchi’s equivalence gives us
the isomorphism

$P\simeq A\,\square_{\pi_{B}}\pi_{B}(P)$

as objects in the category ${}^{A}_{B}\mathrm{Mod}$. Another easy observation
is given by the following lemma, which will be used in the proof of
Proposition 3.10 for the special case of principal pairs.

######  Lemma 3.1.

For $A$ a Hopf algebra and $(B,P)$ a nested pair of quantum homogeneous
$A$-spaces, $P$ is faithfully flat as a right $B$-module.

###### Proof.

Since every object of ${}^{A}_{B}\mathrm{Mod}$ is projective as a right
$B$-module, $P$ is certainly flat as a right $B$-module. Now $P$ is
_faithfully_ flat as a right $B$-module if and only if
$P\otimes_{B}\mathcal{M}\neq 0$, for all left $B$-modules $\mathcal{M}$. If
$P\otimes_{B}\mathcal{M}=0$, then we must have that
$1_{A}\otimes_{B}\mathcal{M}=0$, implying that $A\otimes_{B}\mathcal{M}=0$,
contradicting faithful flatness of $A$ as a right $B$-module. Thus we conclude
that $P$ is faithfully flat as a right $B$-module. ∎

For the special case of $A=P$, a nested pair obviously reduces to a quantum
homogeneous space. The fact that any quantum homogeneous space is a
Hopf–Galois extension of its space of coinvariants is generalised by the
following result.

######  Proposition 3.2.

The map $\mathrm{can}:A\otimes_{B}A\to A\otimes\pi_{B}(A)$ restricts to an
isomorphism

$\displaystyle P\otimes_{B}P\to A\,\square_{\pi_{P}}\pi_{B}(P),$

in ${}^{A}_{P}\mathrm{Mod}$, the category of $(A,P)$-relative Hopf modules.

###### Proof.

Consider the vector space isomorphism:

$\displaystyle\Phi_{P}(P\otimes_{B}P)\simeq\pi_{B}(P),$
$\displaystyle[p^{\prime}\otimes p]\mapsto\varepsilon(p^{\prime})\pi_{B}(p).$

This gives us the commutative diagram

Thus the given map is an isomorphism in ${}^{A}_{B}\mathrm{Mod}$ as claimed. ∎

###  3.2. A Generalisation of Takeuchi’s Equivalence for Nested Pairs

In this subsection we introduce a generalisation of Takeuchi’s equivalence to
the setting of nested pairs. We begin by introducing a generalisation of the
category of relative Hopf modules.

######  Definition 3.3.

For $A$ a Hopf algebra, and $(B,P)$ a nested pair of quantum homogeneous
$A$-spaces, the objects of the category ${}^{A}_{B}\textrm{Mod}^{\pi_{P}}$ are
those objects ${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}$ endowed with a right
$\pi_{P}(A)$-comodule structure
$\Delta_{R,\pi_{P}}:{\mathcal{F}}\to{\mathcal{F}}\otimes\pi_{P}(A)$ giving
${\mathcal{F}}$ the structure of an $(A,\pi_{P}(A))$-bicomodule and such that
$\Delta_{R,\pi_{P}}$ is a left $B$-module map. The morphisms are
$(A,\pi_{P}(A))$-bicomodule and left $B$-module maps.

For any ${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}^{\pi_{P}}$, regarding
${\mathcal{F}}$ as an object in ${}^{A}_{B}\mathrm{Mod}$ gives the left
$\pi_{P}(A)$-comodule $\Phi({\mathcal{F}})$. We introduce a right
$\pi_{P}(A)$-coaction

$\displaystyle\Delta_{R,\pi_{P}}:\Phi({\mathcal{F}})\to\Phi({\mathcal{F}})\otimes\pi_{P}(A),$
$\displaystyle[f]\mapsto[f_{(0)}]\otimes f_{(1)},$

which is well defined since $\Delta_{R,\pi_{P}}$ is by assumption a left
$B$-module map. Operating on morphisms just as for Takeuchi’s equivalence we
get $(\pi_{B}(A),\pi_{P}(A))$-bicomodule maps, and so, we get a functor from
${}^{A}_{B}\mathrm{Mod}^{\pi_{P}}$ to ${}^{\pi_{B}}\mathrm{Mod}^{\pi_{P}}$. By
abuse of notation we again denote this functor by $\Phi$.

In the opposite direction, for any $V\in{}^{\pi_{B}}\mathrm{Mod}^{\pi_{P}}$,
regarding $V$ as an object in ${}^{\pi_{B}}\mathrm{Mod}$ we have the relative
Hopf module $\Psi(V)\in{}^{A}_{B}\mathrm{Mod}$. We can endow $\Psi(V)$ with
the right $\pi_{P}(A)$-coaction

(3)
$\displaystyle(\mathrm{id}\otimes\Delta_{R,\pi_{P}}):\Psi(V)=A\square_{\,\pi_{B}}V\to
A\square_{\pi_{B}}V\otimes\pi_{P}(A),$

which is well defined since $V$ is a $(\pi_{B}(A),\pi_{P}(A))$-bicomdoule.
Operating on morphisms just as for Takeuchi’s equivalence, we get morphisms in
${}^{A}_{B}\mathrm{Mod}$, and so, we have defined a functor from
${}^{\pi_{B}}\mathrm{Mod}^{\pi_{P}}$ to ${}^{A}_{B}\mathrm{Mod}^{\pi_{P}}$. By
abuse of notation we again denote this functor by $\Psi$.

######  Proposition 3.4.

The components of Takeuchi’s natural transformations $\mathrm{U}$ and
$\mathrm{C}$ are right $\pi_{P}(A)$-comodule maps. Hence taken together with
the functors $\Phi$ and $\Psi$, we have an adjoint equivalence between the
categories ${}^{A}_{B}\mathrm{Mod}^{\pi_{P}}$ and
${}^{\pi_{B}}\mathrm{Mod}^{\pi_{P}}$.

###### Proof.

That each component of $\mathrm{U}$ is a $\pi_{P}(A)$-comodule map follows
from the fact that, for any
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}^{\pi_{P}}$, and any
$f\in{\mathcal{F}}$, we have

That the same holds true for $\mathrm{C}$ follows from the fact that, for any
$V\in{}^{\pi_{B}}\mathrm{Mod}^{\pi_{P}}$, and any $\sum_{i}a_{i}\otimes
v_{i}\in A\square_{\,\pi_{B}}V$, we have

Thus $\Phi$, $\Psi$, $\mathrm{U}$, and $\mathrm{C}$ give an adjoint
equivalence as claimed. ∎

######  Remark 3.5.

In [13, Proposition 3.5] an alternative generalisation of Takeuchi’s
equivalence was considered for principal pairs (see Remark 3.11 below). As a
careful reading of the proof will confirm, the argument can be directly
extended to the setting of nested pairs of quantum homogeneous spaces.

###  3.3. Some Consequences of the Equivalence

For any nested pair of quantum homogeneous subspaces $(B,P)$, we have the
surjective Hopf algebra map

$\displaystyle\pi_{B,P}:\pi_{B}(A)\to\pi_{P}(A),$
$\displaystyle\pi_{B}(a)\mapsto\pi_{P}(a).$

Thus $\pi_{P}(A)$ is canonically a quantum subgroup of $\pi_{B}(A)$. This
gives us a right $\pi_{P}(A)$-coaction on $\pi_{P}(A)$, with respect to which
$\pi_{B}(A)$ has the structure of an object in
${}^{\pi_{B}(A)}\mathrm{Mod}^{\pi_{B}(A)}$. The following proposition gives an
easy but important consequence of Takeuchi’s equivalence relating $P$, $B$,
and the space of coinvariants of this right coaction.

######  Proposition 3.6.

For $(B,P)$ a nested pair of quantum homogeneous $A$-spaces:

  1. 1.

an isomorphism in the category ${}^{A}_{B}\mathrm{Mod}$ is given by

$\displaystyle P\to
A\,\square_{\,\pi_{B}}\big{(}\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}\big{)},$
$\displaystyle p\mapsto p_{(1)}\otimes\pi_{B}(p_{(2)}),$

  2. 2.

we have the identity

$\pi_{B}(P)=\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}.$

###### Proof.

1\. The unit of Takeuchi’s equivalence gives a right $\pi_{P}(A)$-comodule
isomorphism $\mathrm{U}:A\to A\,\square_{\,\pi_{B}}\pi_{B}(A)$, thus we see
that

$P=A^{\mathrm{co}(\pi_{P}(A))}\simeq\left(A\,\square_{\,\pi_{B}}\pi_{B}(A)\right)^{\mathrm{co}(\pi_{P}(A))}=A\,\square_{\,\pi_{B}}\left(\pi_{B}(A)\right)^{\mathrm{co}(\pi_{P}(A))}.$

Thus the claimed isomorphism is given by the unit of the equivalence.

2\. Operating by the functor $\Phi$ on the isomorphism given in 1 and then
composing $\Phi(\mathrm{U})$ with the counit $\mathrm{C}$ of the equivalence,
we get the isomorphism

$\displaystyle\mathrm{C}\circ\Phi(\mathrm{U}):\Phi_{B}(P)=\pi_{B}(P)\to\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))},$
$\displaystyle\pi_{B}(p)\mapsto\pi_{B}(p),$

and hence the claimed identity. ∎

######  Remark 3.7.

Motivated by Lemma 3.1, Proposition 3.2, and Proposition 3.6, we think of the
triple of algebras

$B\hookrightarrow P\twoheadrightarrow\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}$

as a _noncommutative homogeneous fibration_ , and refer to it as such, at
least informally. Moreover we refer to $\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}$
as the _fiber_ of the fibration.

###  3.4. Principal Pairs

We now arrive at the main goal of this section: a new and simplified version
of the definition of a principal pair, as first introduced in [13, Definition
3.2]. The precise relationship between the two definitions is discussed in the
remark below. The definition can be viewed as a noncommutative generalisation
of a fibration of homogeneous space over a homogeneous space

$\displaystyle G/H\to G/K,$

where $G$ is a group, and $H\subseteq K\subseteq G$ are subgroups such that
$H$ is normal in $K$. (See [19, §2] for a more detailed discussion.)

######  Definition 3.8.

For a Hopf algebra $A$, a _principal $A$-pair_ is a pair $(B,P)$ of nested
quantum homogeneous $A$-spaces such that
$\pi_{B}(P)=\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}$ is a Hopf subalgebra of
$\pi_{B}(A)$.

We will show how to construct a principal comodule algebra from any principal
pair, beginning with a lemma introducing a right $\pi_{B}(P)$-coaction on $P$.

######  Lemma 3.9.

For a principal $A$-pair $(B,P)$, it holds that

$\Delta_{R,\pi_{B}}(P)\subseteq P\otimes\pi_{B}(P).$

Hence $\Delta_{R,\pi_{B}}$ restricts to a coaction $\Delta_{R,\pi_{B}}:P\to
P\otimes\pi_{B}(P)$ and $B=P^{\mathrm{co}(\pi_{B}(P))}$.

###### Proof.

Let us first show that $P$ is a right $\pi_{B}(A)$-subcomodule of $A$. It
follows from the isomorphism given in Proposition 3.6 that $P$ is a right
$\pi_{B}(A)$-subcomodule of $A$ if and only if
$A\square_{\,\pi_{B}}\pi_{B}(P)$ is a right subcomodule of
$A\square_{\,\pi_{B}}\pi_{B}(A)$. However, this is a direct consequence of our
assumption that $\pi_{B}(P)$ is a Hopf subalgebra of $\pi_{B}(A)$. The fact
that $B=P^{\mathrm{co}(\pi_{B}(P))}$ is evident. ∎

######  Proposition 3.10.

The pair $(P,\Delta_{R,\pi_{B}(P)})$ is a principal comodule algebra.

###### Proof.

It follows from Proposition 3.2 that the canonical map induces an isomorphism

$P\otimes_{B}P\to A\square_{\,\pi_{P}}\pi_{B}(P).$

However, $\pi_{B}(P)$ is a Hopf subalgebra of $\pi_{B}(A)$ by assumption, and
so, it is trivial as a left $\pi_{P}(A)$-comodule, meaning that

$A\square_{\,\pi_{P}}\pi_{B}(P)=A\otimes\pi_{B}(P).$

Thus $\mathrm{can}$ is an isomorphism, which is to say, $P$ is a Hopf–Galois
extension of $B$.

Faithful flatness of $P$ as a right $B$-module follows from Proposition 3.1.
Recalling that $P$ is a Hopf–Galois extension of $B$, it now follows from [38,
Theorem I] that $P$ is also faithfully flat as a left $B$-module. Thus the
pair $(P,\Delta_{R,\pi_{B}(P)})$ is a principal comodule algebra as claimed. ∎

######  Remark 3.11.

From Proposition 3.10 it is clear that any principal $A$-pair $(B,P)$ the pair

$(\pi_{P}:A\to\pi_{P}(A),\pi_{B}:A\to\pi_{B}(A))$

is a principal pair in the original sense of [13, Definition 3.2]. Conversely,
we see that for principal pair in the original sense of [13] will give a
principal pair in the sense of this paper. A subtle point is that the
definition of [13] assumes explicit choices of quantum subgroups $\pi_{H}:A\to
H$ and $\pi_{K}:A\to K$, where no such choice is assumed in this paper.

###  3.5. Schneider’s Equivalence for Principal Pairs

In this subsection we look at Schneider’s equivalence for the special case of
principal pairs. We begin by introducing a new category.

######  Definition 3.12.

For $(B,P)$ a principal pair, the objects of the category
${}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$ are objects ${\mathcal{F}}$ in
${}_{P}\mathrm{Mod}^{\pi_{B}(P)}$ endowed with a left $A$-comodule structure
$\Delta_{L,A}:{\mathcal{F}}\to A\otimes{\mathcal{F}}$ such that
${\mathcal{F}}$ is an $(A,P)$-relative Hopf module, and an
$(A,\pi_{B}(P))$-bicomodule. The morphisms are left $A$-comodule, right
$\pi_{B}(P)$-comodule, left $P$-module maps.

For any ${\mathcal{F}}\in{}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$, regarding
${\mathcal{F}}$ as an object in ${}_{P}\mathrm{Mod}^{\pi_{B}(P)}$, we consider
the object ${\mathcal{F}}^{\mathrm{co}(\pi_{B}(P))}\in{}_{B}\mathrm{Mod}$. We
introduce a left $A$-coaction

$\displaystyle\Delta_{L,A}:{\mathcal{F}}^{\mathrm{co}(\pi_{B}(P))}\to{\mathcal{F}}^{\mathrm{co}(\pi_{B}(P))},$
$\displaystyle f\mapsto f_{(-1)}\otimes f_{(0)},$

which is well defined because ${\mathcal{F}}$ is an
$(A,\pi_{B}(P))$-bicomodule. This gives a functor from
${}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$ to ${}^{A}_{B}\mathrm{Mod}$, operating
on morphisms just as for Schneider’s equivalence, and which by abuse of
notation we again denote by $(-)^{\mathrm{co}(\pi_{B}(P))}$.

In the opposite direction, for any ${\mathcal{N}}\in{}^{A}_{B}\mathrm{Mod}$,
we can endow $P\otimes_{B}{\mathcal{N}}$ with the tensor product left
$A$-coaction, and the right $T$-coaction $\Delta_{R}\otimes\mathrm{id}$. This
gives us a functor

$P\otimes_{B}-:{}^{A}_{B}\mathrm{Mod}\to{}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)},$

which acts on morphisms just as for Schneider’s equivalence.

######  Proposition 3.13.

The functors $(-)^{\mathrm{co}(\pi_{B}(P))}$ and $A\otimes_{B}-$ give an
adjoint equivalence between ${}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$ and
${}^{A}_{B}\mathrm{Mod}$.

###### Proof.

To establish the proposition, we need only show that the unit and counit
morphisms of Schneider’s equivalence are left $A$-comodule maps. For the unit
$\mathrm{U}$ this follows from the commutativity of the following diagram

while for the counit $\mathrm{C}$ we have

Thus the functors $(-)^{\mathrm{co}(\pi_{B}(P))}$ and $A\otimes_{B}-$ give an
adjoint equivalence as claimed. ∎

Combining this with Takeuchi’s equivalence we immediately get the following
corollary, showing that ${}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$ is in fact
equivalent to the category of $\pi_{B}(P)$-comodules.

######  Corollary 3.14.

An equivalence between the categories ${}^{A}_{P}\mathrm{Mod}^{\pi_{B}(P)}$
and ${}^{\pi_{B}}\mathrm{Mod}$ is given by the functors
$\Phi\circ(-)^{\mathrm{co}(\pi_{B}(P))}$ and $(A\otimes_{B}-)\circ\Psi$.

##  4\. Group Algebra Principal Pairs and Relative Line Modules

In this section we introduce a special type of principal pair called a _group
algebra principal pair_. This is motivated by the classical special case of a
homogeneous principal fibration of the form $G/L^{\mathrm{s}}\to G/L$, where
$L$ is a reductive subgroup, and $L^{\mathrm{s}}$ is the semisimple part of
$L$. In this special case, $P$ decomposes into a direct sum of relative line
modules, and moreover, every relative line module over the base $B$ is of this
form.

###  4.1. Group Algebra Principal Pairs

For any Hopf algebra $H$ we denote by $\Gamma_{H}$ the group of grouplike
elements of $H$. Moreover, we denote by $\mathbb{C}\Gamma_{H}$ the group
algebra of $\Gamma_{H}$. Note that $\mathbb{C}\Gamma^{+}_{H}$ is a subalgebra
of $\mathbb{C}\Gamma_{H}$, closed under the antipode, and indeed a two-sided
coideal. Thus the quotient space

$K_{H}:=H/\langle\mathbb{C}\Gamma^{+}_{H}\rangle=H/\langle
g-1\,|\,g\in\Gamma_{H}\rangle$

inherits from $H$ the structure of a Hopf algebra. We denote by
$\mathrm{proj}_{K}:H\to K_{H}$ the canonical Hopf algebra projection.

Let us now specialise to the case where the Hopf algebra $H$ is cosemisimple
with Peter–Weyl decomposition
$H\simeq\bigoplus_{\alpha\in\widehat{H}}H_{\alpha}$. Consider the following
equivalence relation on $\widehat{H}$. For
$\alpha,\,\alpha^{\prime}\in\widehat{H}$, we say that
$\alpha\sim\alpha^{\prime}$ if

$\mathbb{C}\Gamma_{H}H_{\alpha}=\mathbb{C}\Gamma_{H}H_{\alpha^{\prime}}.$

We denote by $\widehat{H}_{\Gamma}$ the associated set of equivalence classes
of elements of $\widehat{H}$, and write

$H_{[\alpha]}:=\mathbb{C}\Gamma_{H}H_{\alpha}.$

We call the coarser decomposition

$H\simeq\bigoplus_{[\alpha]\in\widehat{H}_{\Gamma}}H_{[\alpha]}$

the _$\Gamma$ -decomposition_ of $H$. Note that this is a decomposition of $H$
as left $\mathbb{C}\Gamma^{+}_{H}$ module.

######  Lemma 4.1.

Let $H=\bigoplus_{\alpha\in\widehat{H}}H_{\alpha}$ be a cosemisimple Hopf
algebra, and assume that $\mathbb{C}\Gamma_{H}\subseteq Z_{H}$, where $Z_{H}$
denotes the center of $H$. A direct sum decomposition of $K_{H}$ into
subcoalgebras is given by

(4) $\displaystyle
K_{H}\simeq\bigoplus_{[\alpha]\in\widehat{H}_{\Gamma}}\mathrm{proj}_{K}(H_{[\alpha]})=\bigoplus_{[\alpha]\in\widehat{H}_{\Gamma}}K_{[\alpha]},$

where for each $[\alpha]\in\widehat{H}_{\Gamma},$ we denote
$K_{H_{[\alpha]}}:=\mathrm{proj}_{K}(H_{[\alpha]}).$

###### Proof.

Note first that since $\Gamma_{H}$ is central in $H$, the ideal generated by
$\Gamma_{H}^{+}$ is homogeneous with respect to the $\Gamma$-Peter–Weyl
decomposition, and hence the $\Gamma$-Peter–Weyl decomposition descends to the
quotient. Since, for any $\alpha\in\widehat{H}$, we have

$\mathrm{proj}_{K}(\mathbb{C}\Gamma_{H}H_{\alpha})=\mathrm{proj}_{K}(H_{\alpha}),$

we see that the decomposition is given explicitly by (4). Finally, note that
projection onto the identity component defines a Haar functional proving that
$K_{H}$ is cosemisimple. ∎

######  Theorem 4.2.

Let $A$ and $H$ be Hopf algebras, $\pi_{H}:A\to H$ a surjective Hopf algebra
map, and $\mathrm{proj}_{K}:H\to K_{H}$ the canonical projection. If $H$ is
cosemisimple and $\Gamma_{H}\subseteq Z_{H}$ then

(5)
$\displaystyle\pi_{H}(P)=\pi_{H}(A)^{\mathrm{co}(K_{H})}=\mathbb{C}\Gamma_{H}.$

Moreover, the pair

$\big{(}B:=A^{\mathrm{co}(H)},\,P:=A^{\mathrm{co}(K_{H})}\big{)}$

is a principal $A$-pair.

###### Proof.

Since $H$ is cosemisimple, the space of coinvariants associated to $\pi_{H}$
is a quantum homogeneous space. Moreover, since Lemma 4.1 tells us that
$K_{H}$ is also cosemisimple, the space of coinvariants associated to
$\pi_{K_{H}}$ is also a quantum homogeneous space. Thus the pair $(B,P)$ is a
nested pair of quantum homogeneous $A$-spaces.

Let us now establish (5). The first equality follows directly from Lemma 3.6.
For the second equality, note first that the construction of $K_{H}$ implies
the inclusion

$\displaystyle\Delta_{R,K_{H}}(H_{\alpha})\subseteq H_{\alpha}\otimes
K_{H,\alpha}.$

Thus we see that a non-zero element of $H_{\alpha}$ is right
$K_{H}$-coinvariant if and only if $\alpha$ is equivalent to the trivial
representation. Now since $H^{\mathrm{co}(K_{H})}$ is homogeneous with respect
to the Peter–Weyl decomposition of $H$, we have that
$H^{\mathrm{co}(K_{H})}=\mathbb{C}\Gamma_{H}$. So in particular,
$H^{\mathrm{co}(K_{H})}$ is a Hopf subalgebra of $H$, which is to say, the
nested pair is a principal pair. ∎

We now give a name to the principal pairs considered in the theorem, which we
find convenient to present as a separate definition.

######  Definition 4.3.

Let $A$ be a Hopf algebra, $H$ a cosemisimple Hopf algebra such that
$\Gamma_{H}\subseteq Z_{H}$, and $\pi:A\to H$ a surjective Hopf algebra map.
We call the associated principal pair
$(B=A^{\mathrm{co}(H)},\,P=A^{\mathrm{co}(K_{H})})$ a _group algebra principal
pair_.

###  4.2. Relative Line Modules and Group Algebra Principal Pairs

A $\mathbb{C}\Gamma_{H}$-comodule algebra structure on $P$ is equivalent to a
$\Gamma_{H}$-algebra grading of $P$, which we denote by

(6) $\displaystyle
P\simeq\bigoplus_{\gamma\in\Gamma_{H}}{\mathcal{E}}_{\gamma}.$

######  Proposition 4.4.

For a group algebra principal pair

$\left(B=A^{\mathrm{co}(H)},P=A^{\mathrm{co}(K_{H})}\right)$

we have the following results:

  1. 1.

The $\Gamma_{H}$-grading of $P$ is _strong_ , which is to say,
${\mathcal{E}}_{\gamma}{\mathcal{E}}_{\gamma^{\prime}}={\mathcal{E}}_{\gamma\gamma^{\prime}}$,
for all $\gamma,\gamma^{\prime}\in\Gamma_{H}$.

  2. 2.

Each ${\mathcal{E}}_{\gamma}$ is a relative line module over
$B={\mathcal{E}}_{0}$, in particular
$\Phi({\mathcal{E}}_{\gamma})=\pi_{B}({\mathcal{E}}_{\gamma})=\mathbb{C}\gamma$.

  3. 3.

Every relative line module ${\mathcal{E}}$ over $B$ is isomorphic to
${\mathcal{E}}_{\gamma}$, for some $\gamma\in\Gamma_{H}$.

  4. 4.

The decomposition in (6) can equivalently be described as the unique
decomposition of $P$ into its simple sub-objects.

###### Proof.

1\. Since $P$ is a Hopf–Galois extension of $B$, it necessarily follows that
$P$ is strongly graded (see [30, Theorem 8.1.7]).

2\. It is clear from the construction of ${\mathcal{E}}_{\gamma}$ that
$\Phi({\mathcal{E}}_{\gamma})=\pi_{B}({\mathcal{E}}_{\gamma})=\mathbb{C}\gamma$,
and so, ${\mathcal{E}}_{\gamma}$ is a relative line module.

3\. Every simple $H$-comodule is embeddable into $H$. Since
$\Phi_{B}({\mathcal{E}})$ is a one-dimensional $H$-comodule by assumption, it
is embeddable into $\mathbb{C}\Gamma_{H}$. Thus we must have that an
isomorphism $\Phi({\mathcal{E}})\simeq\Phi({\mathcal{E}}_{\gamma})$, for some
$\gamma\in\Gamma_{H}$, or equivalently it must hold that
${\mathcal{E}}\simeq{\mathcal{E}}_{\gamma}$.

4\. Since each ${\mathcal{E}}_{\gamma}$ is simple, the decomposition is indeed
a decomposition of $P$ into simple subobjects. The relative line modules are
all distinct since $\Phi({\mathcal{E}}_{\gamma})$ is not isomorphic to
$\Phi({\mathcal{E}}_{\gamma^{\prime}})$ when $\gamma\neq\gamma^{\prime}$. Thus
we see the decomposition of $P$ into relative line modules is the unique
decomposition into simple sub-objects. ∎

######  Corollary 4.5.

It holds that $\Phi({\mathcal{E}}_{\alpha})=\mathbb{C}[e]$, if and only if
$e\not\in{\mathcal{E}}_{k}^{+}={\mathcal{E}}_{k}\cap A^{+}$.

###### Proof.

Since the $\Gamma_{H}$-grading is strong, there exists a sum
$\sum_{i}e_{i}\otimes e^{\prime}_{i}$ in
${\mathcal{E}}_{-\gamma}\otimes_{B}{\mathcal{E}}_{\gamma}$ such that
$\sum_{i}e_{i}e_{i}^{\prime}=1,$ for any $\gamma\in\Gamma_{H}$. Thus, for any
$e\in{\mathcal{E}}^{+}_{\gamma}$, we have

$[e]=\sum_{i}[ee_{i}e_{i}^{\prime}]=0,$

where the second equality follows from the fact that $ee_{i}\in B^{+}$.

Consider next an element $e\notin{\mathcal{E}}_{k}^{+}$. The coset $[e]=0$ if
and only only if $e=\sum_{i}b_{i}e_{i}$, for some $b_{i}\in
B^{+},\,e_{i}\in{\mathcal{E}}_{\gamma}$. But in this case

$\varepsilon(e)=\sum_{i}\varepsilon(b_{i})\varepsilon(e_{i})=0,$

contradicting our assumption that $e\not\in{\mathcal{E}}^{+}_{\gamma}$. Thus
$\Phi({\mathcal{E}}_{\gamma})$ is spanned by $[e]$. ∎

######  Example 4.6.

Let $A$ and $H$ be Hopf algebras, and $\pi:A\to H$ a surjective Hopf algebra
map. If $H$ is abelian and satisfies $H=\mathbb{C}\Gamma_{H}$, then $\pi:A\to
H$ clearly satisfies the requirements of Theorem 4.2, and so, we have a group
algebra principal pair. In this case $K_{H}\simeq\mathbb{C}1$, implying that
$A=P^{\mathrm{co}(K_{H})}$, and giving us the strong $\Gamma_{H}$-algebra
grading

$A\simeq\bigoplus_{\gamma\in\Gamma_{H}}{\mathcal{E}}_{\gamma},$

in the category ${}^{A}_{B}\mathrm{Mod}_{0}$. Thus every object in
${}^{A}_{B}\mathrm{Mod}_{0}$ is a direct sum of relative line modules. This is
the situation for the full quantum flag manifolds
$\mathcal{O}_{q}(G/\mathbb{T}^{r})$ presented in §5 below.

######  Example 4.7.

The case where where $\Gamma_{H}=1$ again gives a group algebra principal
pair, but in this case $K_{H}$ is clearly isomorphic to $H$, meaning that

$P=A^{\mathrm{co}(K_{H})}=A^{\mathrm{co}(H)}=B,$

which is to say, the Hopf–Galois extension is trivial.

###  4.3. Schneider’s Equivalence for Group Algebra Principal Pairs

We finish this section with some observations about Schneider’s equivalence in
the special setting of group algebra principal pairs. Recalling that comodules
over $\mathbb{C}\Gamma_{H}$ are the same as graded modules over the group
$\Gamma_{H}$, we see that Schneider’s equivalence gives us

${}_{P}\mathrm{Mod}^{\mathrm{gr}(\Gamma_{H})}\simeq{}_{B}\mathrm{Mod}.$

Moreover, the left $A$-covariant version of Schneider’s equivalence gives us

(7)
$\displaystyle{}^{A}_{P}\mathrm{Mod}^{\mathrm{gr}(\Gamma_{H})}\simeq{}^{A}_{B}\mathrm{Mod}\simeq{}^{K_{H}}\mathrm{Mod},$

which is to say, $\Gamma_{H}$-graded $(A,P)$-relative Hopf modules are
equivalent to $K_{H}$-comodules.

##  5\. Quantum Flag Manifolds

In this section we give a principal pair description of the quantum flag
manifolds. To set notation, we begin by recalling the necessary definitions
and results of the Drinfeld–Jimbo quantum enveloping algebras and quantum
coordinate algebras, as well as the quantum flag manifolds
$\mathcal{O}_{q}(G/L_{S})$ and their associated quantum Poisson homogeneous
spaces $\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ introduced by Dijkhuizen and
Stokman [17]. We then present the pair $\mathcal{O}_{q}(G/L_{S})$ and
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ as a group algebra principal pair.
This result is then in turn used to establish a number of categorical
equivalences, to give a complete description of the relative line modules over
$\mathcal{O}_{q}(G/L_{S})$, and to construct an explicit principal $\ell$-map.

###  5.1. Drinfeld–Jimbo Quantum Groups

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra of
rank $r$. We fix a Cartan subalgebra $\mathfrak{h}$ and choose a set of simple
roots $\Pi=\\{\alpha_{1},\dots,\alpha_{r}\\}$ for the corresponding root
system in $\mathfrak{g}^{*}$, where $\mathfrak{g}^{*}$ denotes the linear dual
of $\mathfrak{g}$. We 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 $(a_{ij})_{ij}$ of
$\mathfrak{g}$ is defined by
$a_{ij}:=\big{(}\alpha_{i}^{\vee},\alpha_{j}\big{)},$ where
$\alpha_{i}^{\vee}:=2\alpha_{i}/(\alpha_{i},\alpha_{i})$.

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
$i=1,\ldots,r$, subject to the relations

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

along with the quantum Serre relations which we omit (see [24, §6.1.2] for
details). The formulae

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

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

###  5.2. Type-$1$ Representations

The set of fundamental weights
$\\{\varpi_{1},\dots,\varpi_{r}\\}\subseteq\mathfrak{h}^{*}$ of $\mathfrak{g}$
are defined by $\big{(}\alpha_{i}^{\vee},\varpi_{j}\big{)}=\delta_{ij}$, for
all $i,j=1,\dots,r$. We denote by ${\mathcal{P}}$ the integral weight lattice
of $\mathfrak{g}$, which is to say the $\mathbb{Z}$-span of the fundamental
weights. Moreover, we denote by ${\mathcal{P}}^{+}$ the cone of dominant
integral weights, which is to say the $\mathbb{Z}_{\geq 0}$-span of the
fundamental weights.

A vector $v\in V$, for any $U_{q}(\mathfrak{g})$-module $V$, is called a
_weight vector_ of _weight_ $\mathrm{wt}(v)\in\mathcal{P}$ if

(8) $\displaystyle K_{i}\triangleright v=q^{(\alpha_{i},\mathrm{wt}(v))}v,$
$\displaystyle\textrm{ for all }i=1,\ldots,r.$

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 $v\in V_{\lambda}$, of weight $\lambda$
satisfying $E_{i}\triangleright v=0,\text{for all $i=1,\ldots,r$.}$ We call
such a vector $v$ a highest weight vector. The vector $v$ is uniquely
determined up to scalar multiple. We call any finite direct sum of such
$U_{q}(\mathfrak{g})$-representations a type-$1$ representation, and we denote
by ${}_{U_{q}(\mathfrak{g})}\mathbf{type}_{1}$ the full subcategory of
${U_{q}(\mathfrak{g})}$-modules whose objects are finite sums of type-1
modules. Each type-$1$ module $V_{\lambda}$ decomposes into a direct sum of
_weight spaces_ , which is to say, those subspaces of $V_{\lambda}$ spanned by
weight vectors. We now choose, once and for all, a weight basis
$\\{v_{i}\\}_{i=1}^{N_{\lambda}}$, for each irreducible representation
$V_{\lambda}$, for $N_{\lambda}:=\dim(V_{\lambda})$, which is to say a vector
space basis of $V_{\lambda}$ whose elements are weight vectors. We label our
basis so that $v_{N_{\lambda}}$ is the highest weight basis vector.

Since $U_{q}(\mathfrak{g})$ has an invertible antipode, we have an equivalence
between the category of left $U_{q}(\mathfrak{g})$-modules and the category of
right $U_{q}(\mathfrak{g})$-modules, induced by the antipode. For any finite-
dimensional left $U_{q}(\mathfrak{g})$-module $V$, we denote by $V^{*}$ the
$\mathbb{C}$-linear dual of $V$, endowed with its right
$U_{q}(\mathfrak{g})$-module structure. With respect to the equivalence of
left and right $U_{q}(\mathfrak{g})$-modules, the left module corresponding to
$V^{*}_{\mu}$ is isomorphic to $V_{-w_{0}(\mu_{S})}$, where $w_{0}$ denotes
the longest element in the Weyl group of $\mathfrak{g}$. When discussing a
specific irreducible representation $V_{\lambda}$, we usually find it
convenient to denote by $\\{f_{i}\\}_{i=1}^{N_{\lambda}}$ the basis of
$V_{-w_{0}(\lambda)}$ dual to $\\{v_{i}\\}_{i=1}^{N_{\lambda}}$.

###  5.3. Quantum Levi Subalgebras

For $S$ a proper subset of simple roots, consider the Hopf subalgebra of
$U_{q}(\mathfrak{g})$ given by

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

We denote $S^{c}:=\Pi\backslash S$, and represent $S^{c}$ graphically by
coloured nodes in the Dynkin diagram of $\mathfrak{g}$. The definition of a
type-$1$ module for $U_{q}(\mathfrak{g})$ has an evident analogue for the
subalgebra $U_{q}(\mathfrak{l}_{S})$, giving us the category
${}_{U_{q}(\mathfrak{l}_{S})}\mathbf{type}_{1}$.

Classically the algebra $\mathfrak{l}_{S}$ is reductive, and hence decomposes
into a direct sum of a semisimple part and a central part. This means that in
the quantum setting we are motivated to consider the subalgebra

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

Just as in the classical case, the sublattice

$\mathcal{P}^{+}_{S}+\mathcal{P}_{S^{c}}\subseteq\mathcal{P},$

labels the one-dimensional type-$1$ representations of
$U_{q}(\mathfrak{l}_{S})$, where we have denoted

$\displaystyle\mathcal{P}^{+}_{S}:=$
$\displaystyle\,\Big{\\{}\lambda\in\mathcal{P}^{+}\,|\,\lambda=\sum^{\,}_{s\in
S}\,a_{s}\varpi_{s},\textrm{ for some }a_{s}\in\mathbb{Z}_{\geq 0}\Big{\\}},$
$\displaystyle\mathcal{P}_{S^{\mathrm{c}}}:=$
$\displaystyle\,\Big{\\{}\lambda\in\mathcal{P}\,|\,\lambda=\sum^{\,}_{x\in
S^{\mathrm{c}}}\,a_{x}\varpi_{x},\textrm{ for some
}a_{x}\in\mathbb{Z}\Big{\\}}.$

Since $\mathfrak{l}_{S}^{\,\mathrm{s}}$ is a semisimple Lie algebra, it admits
no non-trivial one-dimensional representations, which means that the Hopf
algebra $U_{q}(\mathfrak{l}_{S}^{\,\mathrm{s}})$ admits no non-trivial
$1$-dimensional representations. From this it is clear that the one-
dimensional representations of $U_{q}(\mathfrak{l}_{S})$ are labelled by the
sublattice $\mathcal{P}_{S^{c}}$.

###  5.4. Quantum Coordinate Algebras and the Quantum Flag Manifolds

Let $V$ be a finite-dimensional $U_{q}(\mathfrak{g})$-module, $v\in V$, and
$f\in V^{*}$. Consider the function
$c^{\textrm{\tiny$V$}}_{f,v}:U_{q}(\mathfrak{g})\to{\mathbb{C}}$ defined by
$c^{\textrm{\tiny$V$}}_{f,v}(X):=f\big{(}Xv\big{)}$. The coordinate ring of
$V$ is the subspace

$\displaystyle
C(V):=\text{span}_{\mathbb{C}}\\!\left\\{c^{\textrm{\tiny$V$}}_{f,v}\,|\,v\in
V,\,f\in V^{*}\right\\}\subseteq U_{q}(\mathfrak{g})^{\circ}.$

A Hopf subalgebra of $U_{q}(\mathfrak{g})^{\circ}$ is given by

$\displaystyle\mathcal{O}_{q}(G):=\bigoplus_{\mu\in\mathcal{P}^{+}}C(V_{\mu}).$

We call $\mathcal{O}_{q}(G)$ the quantum coordinate algebra of $G$, where $G$
is the compact simply-connected Lie group having $\mathfrak{g}$ as its
complexified Lie algebra. Note that the Hopf algebra $\mathcal{O}_{q}(G)$ is
cosemisimple by construction.

Consider now the coideal subalgebra of $U_{q}(\mathfrak{l}_{S})$-invariants

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

with respect to the natural left $U_{q}(\mathfrak{g})$-module structure on
$\mathcal{O}_{q}(G)$. Just as for $U_{q}(\mathfrak{g})$, we can talk about the
_type-1 dual_ $\mathcal{O}_{q}(L_{S})$ of $U_{q}(\mathfrak{l}_{S})$.
Equivalently, $\mathcal{O}_{q}(G/L_{S})$ can be described as the space of
coinvariants of the right $\mathcal{O}_{q}(L_{S})$-coaction associated to the
Hopf algebra map

$\rho_{S}:\mathcal{O}_{q}(G)\to\mathcal{O}_{q}(L_{S}),$

given by restriction of domains. Indeed, since any finite-dimensional
$U_{q}(\mathfrak{l}_{S})$-module can be embedded into a finite-dimensional
$U_{q}(\mathfrak{g})$-module, this map is surjective, and

$\pi_{\mathcal{O}_{q}(G/L_{S})}\simeq\mathcal{O}_{q}(L_{S}).$

Since $\mathcal{O}_{q}(L_{S})$ is cosemisimple by construction,
$\mathcal{O}_{q}(G/L_{S})$ is a quantum homogeneous space. We call it the
quantum flag manifold associated to $S$.

Since any finite-dimensional $U_{q}(\mathfrak{l}_{S})$-module can be embedded
into a finite-dimensional $U_{q}(\mathfrak{g})$-module,

###  5.5. The Associated Quantum Poisson Homogeneous Spaces

Consider now the coideal subalgebra of
$U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S})$-invariants

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

which we call the _quantum homogeneous Poisson space associated_ to $S$. Just
as for the quantum flag manifolds, we have an isomorphism

$\pi_{\mathcal{O}_{q}(G/L^{\mathrm{\,s}}_{S})}(\mathcal{O}_{q}(G))\simeq\mathcal{O}_{q}(L^{\mathrm{\,s}}_{S}),$

where $\mathcal{O}_{q}(L^{\mathrm{\,s}}_{S})$ is the type-$1$ dual of
$U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S})$. Since
$\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S})$ is cosemisimple by construction,
$\mathcal{O}_{q}(G/L_{S})$ is a quantum homogeneous space.

As shown in [39, §Theorem 4.1], a set of generators for
$\mathcal{O}_{q}(G/L_{S}^{\,\mathrm{s}})$ is given by

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

where we have denoted $N_{x}:=N_{\varpi_{x}}=\dim(V_{\varpi_{x}})$, and
$\\{v_{i}\\}_{i=1}^{N_{x}}$, and $\\{f_{i}\\}_{i=1}^{N_{x}}$, are the weight
bases of $V_{\varpi_{x}}$, and $V_{-w_{0}(\varpi_{x})}$ respectively, chosen
in §5.1. We also find it convenient to introduce special notation for the
following distinguished elements

(10) $\displaystyle z^{\varpi_{x}}:=z^{\varpi_{x}}_{f_{N_{x}}},$
$\displaystyle\overline{z}^{\,\varpi_{x}}:=\overline{z}_{v_{N_{x}}}^{\,\varpi_{x}}.$

The elements $z^{\varpi_{x}}$ and $\overline{z}^{\,\varpi_{x}}$ have many
distinguishing features which merit this separate notation, the most evident
being that they are not contained in $\mathcal{O}_{q}(G)^{+}$.

######  Example 5.1.

Note that $\mathcal{O}_{q}(SU_{2})$ is the quantum Poisson homogeneous space
of the Podleś sphere $\mathcal{O}_{q}(S^{2})$, the unique quantum flag
manifold of $\mathcal{O}_{q}(SU_{2})$. In this case there is just one
fundamental representation $V_{\varpi_{1}}$ and it is of dimension $2$. The
associated generators

$\displaystyle u_{12}:=z^{\varpi_{1}}_{1}=c^{\varpi_{1}}_{12},$ $\displaystyle
u_{22}:=z^{\varpi_{1}}_{2}=c^{\varpi_{1}}_{22},$ $\displaystyle
u_{11}:=\overline{z}^{\varpi_{1}}_{1}=c^{\varpi_{1}}_{11},$ $\displaystyle
u_{21}:=\overline{z}^{\varpi_{1}}_{2}=c^{\varpi_{1}}_{21},$

are just the matrix coefficients of $V_{\varpi_{1}}$, and so, they reduce to
the well-known set of $\mathcal{O}_{q}(SU_{2})$ generators [3, Proposition
2.13].

###  5.6. Quantum Flag Manifolds and Principal Pairs

In this subsection we identify the Hopf algebras $K_{\mathcal{O}_{q}(L_{S})}$
and $\mathcal{O}_{q}(L_{S}^{\mathrm{s}})$ and conclude that the quantum
Poisson homogeneous space and the quantum flag manifold give a principal pair.
We begin with a lemma establishing centrality of the grouplike elements of
$\mathcal{O}_{q}(L_{S})$.

######  Lemma 5.2.

Every element of $\Gamma_{\mathcal{O}_{q}(L_{S})}$ is central in
$\mathcal{O}_{q}(L_{S})$.

###### Proof.

Note first that any $\gamma\in\Gamma_{\mathcal{O}_{q}(L_{S})}$ is a coordinate
function associated to a one-dimensional representation $V_{\lambda}$, for
$\lambda\in\mathcal{P}_{S}$. This means that any element of
$U_{q}(\mathfrak{l}_{S}^{\,\mathrm{s}})$ must act trivially on $V$, and hence
the only generators of $U_{q}(\mathfrak{l}_{S})$ that act non-trivially on $V$
are $K_{x}^{\pm 1}$, for $x\in S^{\mathrm{c}}$. Consider a general monomial
$X$ in the generators $E_{j},F_{j}$, for $j\in S$, then take the general
product

$\displaystyle Y:=(\Pi_{i=1}^{r}K_{i}^{a_{i}})X\in U_{q}(\mathfrak{l}_{S}),$
$\displaystyle\textrm{ for }a_{1},\dots,a_{r}\in\mathbb{Z},$

and note that for all $h\in\mathcal{O}_{q}(L_{S})$

$\displaystyle\gamma h(Y)=\gamma(Y_{(1)})h(Y_{(2)})=\,$
$\displaystyle\gamma\big{(}(\Pi_{i=1}^{r}K_{i}^{a_{i}})X_{(1)}\big{)}h((\Pi_{i=1}^{r}K_{i}^{a_{i}})X_{(2)})$
$\displaystyle=$ $\displaystyle\,\gamma\big{(}\Pi_{x\in
S^{\mathrm{c}}}K_{x}^{a_{x}}\big{)}h((\Pi_{i=1}^{r}K_{i}^{a_{i}})X).$

Similarly it holds that

$h\gamma(Y)=h((\Pi_{i=1}^{r}K_{i}^{a_{i}})X)\gamma\big{(}\Pi_{x\in
S^{\mathrm{c}}}K_{x}^{a_{x}}\big{)}.$

Thus we see that $\gamma$ and $h$ commute for all
$h\in\mathcal{O}_{q}(L_{S})$. ∎

######  Proposition 5.3.

The pair
$\big{(}\mathcal{O}_{q}(G/L_{S}),\,\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\big{)}$
is a group algebra principal pair.

###### Proof.

Recall the Peter–Weyl decompositions

$\displaystyle\mathcal{O}_{q}(L_{S})\simeq\bigoplus_{\lambda+\mu\in\mathcal{P}_{S}^{+}+\mathcal{P}^{\,}_{S^{c}}}C(U_{\lambda+\mu}),$
$\displaystyle\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S})=\bigoplus_{\mu\in\mathcal{P}_{S}^{+}}C(W_{\mu}),$

and consider the coarser decomposition of $\mathcal{O}_{q}(L_{S})$ given by

$\displaystyle\mathcal{O}_{q}(L_{S})\simeq\bigoplus_{\lambda\in\mathcal{P}_{S}}A_{\lambda},$
$\displaystyle\textrm{ where
}A_{\lambda}:=\bigoplus_{\mu\in\mathcal{P}^{{}^{\,}}_{S^{c}}}C(U_{\lambda+\mu}).$

We note that the kernel of the Hopf algebra surjection
$\rho:\mathcal{O}_{q}(L_{S})\twoheadrightarrow\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$
is homogeneous with respect to this coarser decomposition. It holds that

(11) $\displaystyle c^{V_{\lambda}}_{f,v}-1\in\ker(\rho),$
$\displaystyle\textrm{ for all }\lambda\in\mathcal{P}^{+}_{S^{c}},\,f\in
U_{\lambda}^{*},\,v\in U_{\lambda},$

and moreover, the ideal $I$ generated by these elements is homogeneous with
respect to the coarser grading, giving us a decomposition

$I\simeq\bigoplus_{\lambda\in\mathcal{P}^{+}_{S}}I_{\lambda}.$

Now the quotient $A_{\lambda}/I_{\lambda}$ has dimension less than or equal to
$C(W_{\lambda})$, which together with surjectivity of $\rho$ implies that the
dimensions are equal, which is to say, the kernel of $\rho$ is generated by
the elements given in (11). Thus we see that
$\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S})$ is isomorphic as a Hopf algebra to
$K_{\mathcal{O}_{q}(L_{S})}$. Thus, since $\mathcal{O}_{q}(L_{S})$ is
cosemisimple by construction, and we have shown in the previous lemma that all
the grouplike elements of $\mathcal{O}_{q}(L_{S})$ are central, the given pair
is a principal pair. ∎

######  Corollary 5.4.

The categorical equivalence

$\displaystyle{}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\,\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}^{\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S})}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\
{}^{\mathcal{O}_{q}(L_{S})}\mathrm{Mod}^{{\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S}})}.$

follows immediately from Proposition 3.4.

######  Example 5.5.

For any Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{g})$, the quantum flag
manifold $\mathcal{O}_{q}(F_{G})$ associated to the choice of simple nodes
$S=\varnothing$ is called the _full quantum flag manifold_ of
$\mathcal{O}_{q}(G)$. In this case the associated Hopf algebra
$\mathcal{O}_{q}(L_{S})$ is isomorphic to $\mathcal{O}(\mathbb{T}^{r})$, which
is to say the group Hopf algebra of $\mathbb{Z}^{r}$. This gives a strong
$\mathbb{Z}^{r}$-algebra grading

$\mathcal{O}_{q}(G)\simeq\bigoplus_{\gamma\in\mathbb{Z}^{r}}{\mathcal{E}}_{\gamma}.$

In other words, we have a decomposition of $\mathcal{O}_{q}(G)$ into relative
line modules. Since $\mathcal{O}(\mathbb{T}^{r})$ is spanned by grouplike
elements, this can be see as a degenerate case of Theorem 5.3 in the spirit of
Example 4.6.

######  Example 5.6.

The full quantum flag manifold of $\mathcal{O}_{q}(SU_{2})$ is the Podleś
sphere, and the associated algebra grading of $\mathcal{O}_{q}(SU_{2})$ is a
$\mathbb{Z}$-grading. Recalling the notation of Example 5.1, the grading is
determined by

$\displaystyle\mathrm{deg}(u_{11})=\mathrm{deg}(u_{21})=-1,$
$\displaystyle\mathrm{deg}(u_{12})=\mathrm{deg}(u_{22})=1.$

Thus we recover the well-known $\mathbb{Z}$-grading of
$\mathcal{O}_{q}(SU_{2})$, as considered, for example, in [27, §1] up to a
change of sign.

###  5.7. Relative Line Modules

Since
$\big{(}\mathcal{O}_{q}(G/L_{S}),\,\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\big{)}$
is a principal pair, Lemma 4.4 implies the following result.

######  Proposition 5.7.

The direct sum decomposition of $\mathcal{O}_{q}(G/L_{S}^{\,\mathrm{s}})$ into
simple subobjects is a decomposition into relative line modules

$\mathcal{O}_{q}(G/L_{S}^{\,\mathrm{s}})\simeq\bigoplus_{\gamma\in\mathcal{P}_{S^{c}}}{\mathcal{E}}_{\gamma}.$

This gives $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ the structure of a
strongly $\mathcal{P}_{S^{c}}$-graded algebra. Moreover, every relative line
module over $\mathcal{O}_{q}(G/L_{S})$ is isomorphic to
${\mathcal{E}}_{\gamma}$, for some $\gamma\in\mathcal{P}_{S^{\mathrm{c}}}$.

We now give a concrete description of the decomposition of
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ into line bundles in terms of the
generating set of $\mathcal{O}_{q}(G/L^{\,\mathrm{s}})$ presented in (9).

######  Corollary 5.8.

The $\mathcal{P}_{S^{c}}$-grading of $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$
is determined by

$\displaystyle\mathrm{deg}(z^{\varpi_{x}}_{i})=\varpi_{x},$
$\displaystyle\mathrm{deg}(\overline{z}^{\,\varpi_{x}}_{i})=-\varpi_{x},$

where $i=1,\dots,N_{x}$, and $x\in S^{c}$.

###### Proof.

The right $\mathcal{O}_{q}(L_{S})$-coaction on
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ acts on $z^{\varpi_{x}}_{i}$
according to

$\displaystyle\Delta_{R,\mathcal{O}_{q}(L_{S})}(z^{\varpi_{x}}_{i})=\sum_{a=1}^{N_{x}}c^{V_{\varpi_{x}}}_{v_{i},f_{a}}\otimes\pi_{\mathcal{O}_{q}(L_{S})}(\,z^{\varpi_{x}}_{a}).$

It follows from (5) in Theorem 4.2 that
$\pi_{\mathcal{O}_{q}(L_{S})}(z^{\varpi_{x}})$ is an element of
$\Gamma_{\mathcal{O}_{q}(L_{S})}$. Moreover, for any $y\in S^{c}$,

$\displaystyle\langle
K_{y},\pi_{\mathcal{O}_{q}(L_{S})}(\,z^{\varpi_{x}}_{a})\rangle=\langle
K_{y},z^{\varpi_{x}}_{a}\rangle=f_{a}(K_{y}\triangleright
N_{x})=q^{(\alpha_{y},\varpi_{x})}\delta_{a,N_{x}},$ $\displaystyle\textrm{
for all }a=1,\dots,N_{x}.$

Thus we see that

$\displaystyle\Delta_{R,\mathcal{O}_{q}(L_{S})}(z^{\varpi_{x}}_{i})\in\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})\otimes
C(U_{\varpi_{x}}),$

which is to say, $z^{\varpi_{x}}_{i}$ is of degree $\varpi_{x}$. An analogous
calculation shows that the degree of $\overline{z}^{\,\varpi_{x}}_{i}$ is
$-\varpi_{x}$. Finally, since the $\mathcal{P}_{S^{\mathrm{c}}}$-grading is an
algebra grading, it is completely determined by the degrees of the generators.
∎

###  5.8. A Principal $\ell$-Map

We now produce an explicit principal $\ell$-map for our principal comodule
algebra. This construction generalises the principal $\ell$-map constructed
for the quantum Grassmannians in [13, Corollary 4.13]. For the special case of
the Podleś sphere, the $\ell$-map reduces to the well-known $q$-monopole
connection introduced in [8, 9, 20].

In the statement of the proposition we find the following notation useful

(12) $\displaystyle z_{\gamma}:=\prod_{x\in
S^{\mathrm{c}}}(\widecheck{z}^{\,\varpi_{x}})^{a_{x}},$ $\displaystyle\textrm{
for }\gamma=\sum_{x\in
S^{\mathrm{c}}}a_{x}\varpi_{x}\in\mathcal{P}_{S^{\mathrm{c}}},$

where we have denoted

$\displaystyle(\widecheck{z}^{\,\varpi_{x}})^{a_{x}}:=(z^{\varpi_{x}})^{a_{x}},\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{ if }a_{x}\geq 0,$ and
$\displaystyle(\widecheck{z}^{\,\varpi_{x}})^{a_{x}}:=(\overline{z}^{\,\varpi_{x}})^{a_{x}},\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{ if }a_{x}<0.$

Moreover, we find it convenient to denote

$\displaystyle
t_{\gamma}:=t_{1}^{a_{1}}\cdots\,t_{|S^{\mathrm{c}}|}^{a_{|S^{\mathrm{c}}|}},$
$\displaystyle\textrm{ for any }\gamma=\sum_{x\in
S^{\mathrm{c}}}a_{x}\varpi_{x}\in\mathcal{P}_{S^{\mathrm{c}}}.$

######  Proposition 5.9.

For the Hopf–Galois extension
$\mathcal{O}_{q}(G/L_{S})\hookrightarrow\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$,
a principal $\ell$-map is given by

$\displaystyle\ell:\mathbb{C}\mathcal{P}_{S^{c}}\to\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})\otimes\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}}),$
$\displaystyle t_{\gamma}\mapsto(S\otimes\mathrm{id})\circ\Delta(z_{\gamma}).$

###### Proof.

Let us first check that $\ell$ is well defined. Note first that since $\Delta$
is an algebra map, and $S$ is an anti-algebra map, it is enough to check that
$\ell(z^{\varpi_{x}})\in\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})\otimes\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$,
for each $x\in\mathcal{P}_{S^{c}}$. Since

$\displaystyle\ell(z^{\varpi_{x}})=\sum_{a=1}^{N_{x}}S(z^{\varpi_{x}}_{f_{N_{x}},a})\otimes
z^{\varpi_{x}}_{a,f_{N_{x}}},$

this amounts to showing that
$S(z^{\varpi_{x}}_{f_{N_{x}},a})\in\mathcal{O}_{q}(G/L_{S}^{\mathrm{s}})$, but
this follows from the fact that

$\displaystyle X\triangleright
S(z^{\varpi_{x}}_{f_{N_{x}},a})=S(z^{\varpi_{x}}_{X\triangleright
f_{N_{x}},a})=\varepsilon(X)S(z^{\varpi_{x}}_{f_{N_{x}},a}),$
$\displaystyle\textrm{ for all }X\in U_{q}(\mathfrak{l}_{S}).$

Thus $\ell$ is indeed well defined.

By definition it holds that
$\ell(1_{\mathbb{C}\mathcal{P}_{S^{c}}})=1_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}$.
Moreover, it is clear that

$m_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\circ\ell=\varepsilon
1_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}.$

Right $\mathbb{C}\mathcal{P}_{S^{c}}$-covariance of $\ell$ follows from the
calculation

$\displaystyle(\mathrm{id}\otimes\Delta_{R})\circ\ell(t_{\gamma})=$
$\displaystyle\,(\mathrm{id}\otimes\Delta_{R})\circ(S\otimes\mathrm{id})\circ\Delta(z_{\gamma})$
$\displaystyle=$
$\displaystyle\,(S\otimes\mathrm{id}\otimes\mathrm{id})\circ(\Delta\otimes\mathrm{id})\circ(\Delta_{R})(z_{\gamma})$
$\displaystyle=$
$\displaystyle\,(S\otimes\mathrm{id}\otimes\mathrm{id})\circ(\Delta\otimes\mathrm{id})(z_{\gamma}\otimes
t_{\gamma})$ $\displaystyle=$
$\displaystyle\,((S\otimes\mathrm{id})\circ\Delta(z_{\gamma}))\otimes
t_{\gamma}$ $\displaystyle=$ $\displaystyle\,\ell(t_{\gamma})\otimes
t_{\gamma}$ $\displaystyle=$
$\displaystyle\,(\ell\otimes\mathrm{id})\circ\Delta(t_{\gamma}).$

Left $\mathbb{C}\mathcal{P}_{S^{c}}$-covariance of $\ell$ follows analogously,
and so, we have a principal $\ell$-map. ∎

######  Remark 5.10.

As explained in Appendix A.4, associated to the principal $\ell$-map
constructed above we have a principal connection $\Pi_{\ell}$ acting on the
universal calculus of $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$. As is easily
checked, the map $\Pi_{\ell}$ is a left $\mathcal{O}_{q}(G)$-comodule map,
meaning that every associated relative Hopf module connection $\nabla$ is also
an $\mathcal{O}_{q}(G)$-comodule map. This extends the situation for the
quantum Grassmannians discussed in [13, Corollary 4.13].

###  5.9. Non-Cleftness of the Hopf–Galois Extension

Let us recall the definition of a cleft comodule algebra, which can be viewed
as a noncommutative generalisation of a trivial bundle: For $H$ a Hopf
algebra, we say that a right $H$-comodule algebra $(P,\Delta_{R,H})$ is
_cleft_ if there exists a convolution invertible right $H$-comodule map
$j:H\to P$, which is necessarily injective. We call such a $j$ a _cleaving
map_. If $(P,\Delta_{R,H})$ is a Hopf–Galois extension of
$B=P^{\mathrm{co}(H)}$, then the existence of a cleaving map is equivalent to
$A$ having the _normal basis property_ , which is to say, equivalent to the
existence of a left $B$-module and right $H$-comodule isomorphism $A\simeq
B\otimes H$, see [5, Proposition 2.3] for details.

For the special case of the Podleś sphere, it was shown in [20] that the
Hopf–Galois extension
$\mathcal{O}_{q}(S^{2})\hookrightarrow\mathcal{O}_{q}(SU_{2})$ is not cleft.
We now extend this result to include all quantum flag manifolds.

######  Proposition 5.11.

The Hopf–Galois extension
$\mathcal{O}_{q}(G/L_{S})\hookrightarrow\mathcal{O}_{q}(G/L_{S}^{\,\mathrm{s}})$
is non-cleft.

###### Proof.

Let us assume that we have a convolution invertible right
$\mathbb{C}\mathcal{P}_{S}$-comodule map

$j:\mathbb{C}\mathcal{P}_{S}\to\mathcal{O}_{q}(G/L^{\,s}_{S}).$

Choose an element $x\in S^{c}$. The fact that $j$ is convolution invertible,
and that $t_{x}$ is grouplike, means that $j(t_{x})$ is invertible. Moreover,
since $j$ is a $\mathbb{C}\mathcal{P}_{S}$-comodule map, $j(t_{x})$ must have
degree $\varpi_{x}$.

Consider the associated inclusion
$i_{x}:U_{q}(\mathfrak{sl}_{2})\hookrightarrow U_{q}(\mathfrak{g})$, along
with the dual Hopf algebra surjection
$\pi_{x}:\mathcal{O}_{q}(G)\to\mathcal{O}_{q}(SU_{2})$. Taking $K\in
U_{q}(\mathfrak{sl}_{2})$, we see that

$\displaystyle
K\triangleright\pi_{x}(j(t_{x}))=\,\pi_{x}(j(t_{x})_{(1)})\langle
K_{x},j(t_{x})_{(2)}\rangle=\,q^{(\alpha_{x},\alpha_{x})/2}\pi_{x}(j(t_{x})).$

Thus, with respect to the $\mathbb{Z}$-grading on $\mathcal{O}_{q}(SU_{2})$
induced by the action of the generators $K$ and $K^{-1}$, the element
$\pi_{x}(j(t_{x}))$ has non-zero degree.

Since $\pi_{x}$ is an algebra map, the element $\pi_{x}(j(t_{x}))$ is
necessarily invertible in $\mathcal{O}_{q}(SU_{2})$. It was established in
[20, Appendix] that the only invertible elements in $\mathcal{O}_{q}(SU_{2})$
are scalar multiples of the identity. However, this contradicts the fact that
$\pi_{x}(j(t_{x}))$ has non-zero degree. Thus we conclude that no such $j$
exists, which is to say, that the Hopf–Galois extension is non-cleft. ∎

###  5.10. Schneider’s Equivalence and a Family of Examples

Since $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ and $\mathcal{O}_{q}(G/L_{S})$
form a principal pair, the equivalence

${}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\mathrm{Mod}^{\mathrm{gr}(\mathcal{P}_{S^{\mathrm{c}}})}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ {}^{\mathcal{O}_{q}(L_{S})}\mathrm{Mod}$

follows immediately from (7). In this subsection, we present a naturally
occurring family of objects, in the left hand side category, coming from
quantum group noncommutative geometry.

A quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$ is said to be _irreducible_
if $S^{\mathrm{c}}$ contains a single root vector $\alpha_{x}$ and this root
vector appears in any positive root of $\mathfrak{g}$ with coefficient at most
one. In [21, 22] Heckenberger and Kolb introduced a left
$\mathcal{O}_{q}(G)$-covariant first-order differential calculus
$\Omega^{1}_{q}(G/L_{S})$ for each irreducible quantum flag manifold
$\mathcal{O}_{q}(G/L_{S})$. These calculi are of classical dimension and
$q$-deform the classical space of $1$-forms over $G/L_{S}$. These celebrated
calculi are objects of central importance for the noncommutative geometry of
quantum groups, and are naturally objects in the category of relative Hopf
modules ${}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L_{S})}\mathrm{Mod}$.

For the irreducible case, the weight sublattice $\mathcal{P}_{S^{\mathrm{c}}}$
is isomorphic to $\mathbb{Z}$, and so, the category ${}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\mathrm{Mod}^{\mathrm{gr}(\mathcal{P}_{S^{\mathrm{c}}})}$
specialises to the category

${}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\mathrm{Mod}^{\mathrm{gr}(\mathbb{Z})}.$

In [22, §3.2] Heckenberger and Kolb constructed left
$\mathcal{O}_{q}(G)$-covariant first-order calculi
$\Omega^{1}_{q}(G/L^{\,\mathrm{s}}_{S})$ for the associated quantum Poisson
homogeneous spaces $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$. These calculi
are naturally objects in the category ${}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\mathrm{Mod}$.
Moreover, as shown in [12], each calculus is homogeneous with respect to the
$\mathbb{Z}$-grading on $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$, and so,

$\Omega^{1}_{q}(G/L^{\,\mathrm{s}}_{S})\in{}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{q}(G)}_{\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})}\mathrm{Mod}^{\mathrm{gr}(\mathbb{Z})}.$

It then follows from [13, Proposition 3.10] that
$\Omega^{1}_{q}(G/L^{\,\mathrm{s}}_{S})$ gives a quantum principal bundle over
the base $\mathcal{O}_{q}(G/L_{S})$. This is a key step in the proof of the
Borel–Weil theorem for the irreducible quantum flag manifolds given in [12].

##  6\. Examples of Non-Principal Noncommutative Fibrations

In this section we construct two families of non-principal noncommutative
fibrations, and realise many well-known examples of quantum homogeneous spaces
as noncommutative fibers.

###  6.1. Non-Principal Examples with Irreducible Quantum Flag Manifold Bases

For any Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{g})$, and a pair of
subsets $S_{P}\subseteq S_{B}\subseteq\Pi$, we have the triple of algebras

$\mathcal{O}_{q}(G/L_{S_{B}})\hookrightarrow\mathcal{O}_{q}(G/L_{S_{P}})\twoheadrightarrow\mathcal{O}_{q}(L_{S_{P}}/L_{S_{B}}),$

where we have denoted

$\mathcal{O}_{q}(L_{S_{B}}/L_{S_{P}}):=\mathcal{O}_{q}(L_{S_{B}})^{\mathrm{co}\left(\mathcal{O}_{q}(L_{S_{P}})\right)}.$

As discussed in Remark 3.7, we think of this triple as a type of
noncommutative homogeneous fibration. For special cases we get fibrations
built from well-known examples of quantum homogeneous spaces. Here we present
a series of special cases where the base of the fibration is an irreducible
quantum flag manifold, as discussed in §5.10. Throughout we use Humphrey’s
numbering of the Dynkin nodes [23, §11.4] of $\mathfrak{g}$.

Note that for any two subsets $S_{B}\subseteq S_{P}\subseteq\Pi$, we have an
obvious algebra isomorphism

$\mathcal{O}_{q}(L_{S_{B}}/L_{S_{P}})\simeq\mathcal{O}_{q}(L^{\mathrm{s}}_{S_{B}})^{\mathrm{co}\left(\mathcal{O}_{q}(L_{S_{P}\cap
S_{B}})\right)}.$

In fact, this is an isomorphism of
$\mathcal{O}_{q}(L^{\mathrm{\,s}}_{S_{B}})$-comodules. A motivating example
here is the presentation of $\mathcal{O}_{q}(\mathbb{CP}^{n})$ as a quantum
homogeneous $\mathcal{O}_{q}(U_{n+1})$-space. We will tacitly use this
isomorphism in the examples that follow.

######  Example 6.1.

We begin with the example introduced by Brzeziński and Szymański [29] as a
motivating example for a proposed theory of _noncommutative fiber bundles_
[11]. For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sl}_{3})$, we
choose $S^{c}_{B}=\\{\alpha_{2}\\}$ and
$S^{c}_{P}=\\{\alpha_{1},\alpha_{2}\\},$ which we represent respectively with
the coloured Dynkin diagrams

and .

This gives us the noncommutative fibration

$\mathcal{O}_{q}(\mathbb{CP}^{2})\hookrightarrow\mathcal{O}_{q}(F_{SU_{3}})\twoheadrightarrow\mathcal{O}_{q}(S^{2}),$

where $\mathcal{O}_{q}(\mathbb{CP}^{2})$ is the quantum projective plane, and
$\mathcal{O}_{q}(S^{2})=\mathcal{O}_{q}(\mathbb{CP}^{1})$ is the standard
Podleś sphere.

We now generalise this construction to higher orders. There are two natural
ways to do this. First we consider the case where the fiber and the total
space are higher order full quantum flag manifolds, where we note that the
Podleś sphere is in fact the full quantum flag manifold of
$\mathcal{O}_{q}(SU_{2})$.

######  Example 6.2.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sl}_{n+1})$, we choose
$S^{c}_{B}=\\{\alpha_{n}\\}$ and $S^{c}_{P}=\Pi$ which we represent
graphically with the respective coloured Dynkin diagrams

and .

This gives us the fibration

$\mathcal{O}_{q}(\mathbb{CP}^{n})\hookrightarrow\mathcal{O}_{q}(F_{SU_{n+1}})\twoheadrightarrow\mathcal{O}_{q}(F_{SU_{n}}),$

where $\mathcal{O}_{q}(\mathbb{CP}^{n})$ is quantum projective $n$-space.

We now consider an alternative generalisation of Brzeziński and Szymański’s
fibration, where both the base and the fiber are higher order quantum
projective spaces.

######  Example 6.3.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sl}_{n+1})$, we choose
$S^{c}_{B}=\\{\alpha_{n}\\}$ and $S^{c}_{P}=\\{\alpha_{1},\alpha_{n}\\},$
which we represent graphically with the respective coloured Dynkin diagrams

and .

This gives us the fibration

$\mathcal{O}_{q}(\mathbb{CP}^{n})\hookrightarrow\mathcal{O}_{q}\big{(}SU_{n+1}/(U_{n-1}\times
U_{1})\big{)}\twoheadrightarrow\mathcal{O}_{q}(\mathbb{CP}^{n-1}),$

where we note that for $n=2$, the quantum flag manifold
$\mathcal{O}_{q}\big{(}SU_{n+1}/U_{n-1}\times U_{1}\big{)}$ reduces to the
full quantum flag manifold of $\mathcal{O}_{q}(SU_{3})$.

There is no need to confine our attention to the $A$-series, so let us
consider a $C$-series example.

######  Example 6.4.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sp}_{2n})$, we choose
$S^{c}_{B}=\\{\alpha_{n}\\}$ and $S^{c}_{P}=\Pi$ which we represent
graphically with the respective coloured Dynkin diagrams

and .

This gives us the noncommutative fibration

$\mathcal{O}_{q}(\mathbf{L}_{n})\hookrightarrow\mathcal{O}_{q}(F_{Sp_{n}})\to\mathcal{O}_{q}(F_{SU_{n}}),$

where $\mathcal{O}_{q}(\mathbf{L}_{n})$ is the quantum Lagrangian
Grassmannian. In the commutative case, $\mathbf{L}_{n}$ is the space of
complex structures on $\mathbb{H}^{n}$ compatible with the canonical inner
product.

Let us next consider an example from the exceptional Drinfeld–Jimbo quantum
groups.

######  Example 6.5.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{e}_{6})$, we choose
$S^{c}_{B}=\\{\alpha_{6}\\}$ and $S^{c}_{P}=\\{\alpha_{5},\,\alpha_{6}\\},$
which we represent graphically with the respective coloured Dynkin diagrams

and .

This gives us the quantum homogeneous fibration

$\mathcal{O}_{q}(\mathbb{OP}^{2})\hookrightarrow\mathcal{O}_{q}(E_{6}/(SO_{10}\times
U_{1}))\twoheadrightarrow\mathcal{O}_{q}(\mathbf{S}_{5}),$

where $\mathcal{O}_{q}(\mathbb{OP}^{2})$ is the quantum Caley plane, that is,
the irreducible quantum flag manifold of $\mathcal{O}_{q}(E_{6})$, and
$\mathcal{O}_{q}(\mathbf{S}_{5})$ is the quantum spinor variety, one of the
two $D_{5}$ irreducible quantum flag manifolds. In the classical case
$\mathbf{S}_{5}$ is the space of orthogonal complex structures on
$\mathbb{R}^{10}$.

###  6.2. Noncommutative Sphere Fibrations and Quantum Stiefel Manifolds

In this subsection we generalise a different aspect of Brzeziński and
Szymański’s noncommutative fibration, producing fibrations with higher order
quantum spheres as _noncommutative fibers_. Note first that, for any complex
semisimple Lie algebra $\mathfrak{g}$ and any two subsets $S_{P}\subseteq
S_{B}\subseteq\Pi$, a quantum homogeneous fibration is given by the triple

$\mathcal{O}_{q}(G/L_{S_{B}}^{\mathrm{s}})\hookrightarrow\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S_{P}})\twoheadrightarrow\mathcal{O}_{q}(L^{\mathrm{s}}_{S_{B}}/L^{\mathrm{s}}_{S_{P}}),$

where we have denoted by
$\mathcal{O}_{q}(L^{\mathrm{s}}_{S_{B}}/L^{\mathrm{s}}_{S_{P}})$ the space of
$\mathcal{O}_{q}(L^{\,\mathrm{s}}_{S_{P}})$-coinvariants of
$\mathcal{O}_{q}(L^{\mathrm{s}}_{S_{B}})$. As we will see, for special cases
we get fibrations built from well-known examples of quantum homogeneous
spaces.

With respect to Humphrey’s numbering of the Dynkin nodes [23, §11.4], for any
non-exceptional simple Lie algebra $\mathfrak{g}$, we denote

$\displaystyle S_{>m}:=\\{\alpha_{m+1},\dots,\alpha_{\mathrm{n}}\\}.$

Associated to the Hopf subalgebra
$U_{q}(\mathfrak{l}^{\mathrm{\,s}}_{S_{>m}})$ and its quantum subgroup
$\mathcal{O}_{q}(L_{S_{>m}})$, we have the quantum homogeneous
$\mathcal{O}_{q}(G)$-space of $\mathcal{O}_{q}(L_{S_{>m}})$-coinvariants,
which we call the _quantum $m$-Stiefel manifold_ of $\mathcal{O}_{q}(G)$.

######  Example 6.6.

It is instructive to observe that each Stiefel manifold is the total space of
a principle comodule algebra. Indeed, by the results of §5.6, the quantum
Stiefel manifold associated to any $S_{>m}$ is a principal
$\mathbb{C}\mathcal{P}_{S^{c}_{>m}}$-comodule algebra over the quantum flag
manifold $\mathcal{O}_{q}(G/L_{S_{>m}})$.

Let us now place some examples of quantum Stiefel manifolds into $q$-deformed
non-principal fibrations.

######  Example 6.7.

Consider the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sl}_{n+1})$, for
$n\geq 2$, and the subset $S_{>3}$ of simple nodes, for $m=1,\dots,n$. In this
case the quantum Stiefel manifolds reduce to the quantum complex Stiefel
manifolds $\mathcal{O}_{q}(V_{p}\mathbb{C}^{n+1})$ introduced by Podkolzin and
Vainerman [35], and further studied in [37]. Note that for the degenerate case
of $m=1$, we get back the quantum spheres $\mathcal{O}_{q}(S^{2n-1})$ of
Vaksman and Soibelman [42].

Choosing $S_{B}=S_{>1}$ and $S_{P}=S_{>2}$, which we represent graphically
with the respective coloured Dynkin diagrams

and ,

we get the quantum homogeneous fibration

$\mathcal{O}_{q}(S^{2n+1})\hookrightarrow\mathcal{O}_{q}(V_{2}\mathbb{C}^{n+1})\twoheadrightarrow\mathcal{O}_{q}(S^{2n-1}).$

Note that for the special case of $n=3$, the fiber reduces to
$\mathcal{O}_{q}(S^{3})=\mathcal{O}_{q}(SU_{2})$, so the fibration is actually
principal.

######  Example 6.8.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sp}_{2n})$, we choose
$S_{B}=S_{>1}$ and $S_{P}=S_{>3},$ which we represent graphically with the
respective coloured Dynkin diagrams

and .

The Stiefel manifold corresponding to $S_{>1}$ is a $q$-deformation of the
$C$-series presentation of the sphere $S^{4n-1}$ as the homogeneous sphere

$S^{4n-1}\simeq Sp_{2n}/Sp_{2(n-1)}.$

To differentiate it from the $A$-series quantum sphere, we denote it by
$\mathcal{O}_{q}(S^{4n-1}_{\mathbb{H}})$ and call it the _quantum quaternionic
sphere_. The quantum quaternionic sphere has appeared a number of times in the
literature. It was studied in [26] and [10, 11, 29] as the total space of a
noncommutative instanton fibration, where it was called the _quantum
symplectic sphere_. Moreover, the representations and $K$-theory of its
$C^{*}$-algebra completion were examined in [36].

The Stiefel manifold corresponding to $S_{>2}$ is a $q$-deformation of the
coordinate algebra of the classical quaternionic Stiefel manifold
$V_{2}\mathbb{H}^{n}$. Hence we call it the _quantum quaternionic $2$-plane
Stiefel manifold_ and denote it by $\mathcal{O}_{q}(V_{2}\mathbb{H}^{n})$.

Our choice of $S_{B}$ and $S_{P}$ gives the quantum homogeneous fibration

$\mathcal{O}_{q}(S^{4n-1}_{\mathbb{H}})\hookrightarrow\mathcal{O}_{q}(V_{2}\mathbb{H}^{n})\twoheadrightarrow\mathcal{O}_{q}(S_{\mathbb{H}}^{4n-5}).$

Note that for the degenerate case of $n=2$, the quantum sphere
$\mathcal{O}_{q}(S_{\mathbb{H}}^{4n-5})$ reduces to the Hopf algebra
$\mathcal{O}_{q}(S^{3})=\mathcal{O}_{q}(SU_{2})$, and so, the fibration is
principal.

##  Appendix A Some Comments on Quantum Homogeneous Spaces

In this paper we use a definition of a quantum homogeneous space that is
different (although equivalent) to the one used in our previous works. Thus we
spend some time discussing the various formulations of the definition.
Motivated by the important role relative line modules play in the paper, we
also discuss some results about simple relative Hopf modules.

###  A.1. Quantum Homogeneous Spaces as Principal Comodule Algebras

For any quantum homogeneous space $B=A^{\mathrm{co}(\pi_{B}(A))}$, it is well-
known that $A$ is automatically a Hopf–Galois extension of $B$, see for
example [40, Lemma 3.9]. Thus it follows from [31, Theorem 1] that $A$ is also
faithfully flat as a left $B$-module, and hence that it is a principal
comodule algebra.

Cosemisimplicity allows us to verify the faithful flatness condition for a
general coideal subalgebra $B\subseteq A$ satisfying $B^{+}A=AB^{+}$.
Explicitly, if the quotient Hopf algebra $A/B^{+}A$ is cosemisimple, then $A$
is automatically faithfully flat as a right $B$-module, see[16, §3.3] for a
detailed discussion.

###  A.2. Quantum Homogeneous Spaces, Ideals, and Quantum Subgroups

In the literature it is common to consider a weaker version of the quantum
homogeneous space definition: coideal subalgebras $B\subseteq A$ for which $A$
is faithfully flat as a right $B$-module, but for which $AB^{+}$ is not
necessarily equal to $B^{+}A$. As shown in [28, Theorem 1.11], these
_generalised quantum homogeneous spaces_ are in bijective correspondence with
two-sided coideals $I\subseteq A$ that are also right $A$-ideals and for which
$A$ is faithfully coflat as a right $A/I$-comodule. Explicitly, the
correspondence is given by associating a generalised quantum homogeneous space
$B$ to the right ideal two-sided coideal $B^{+}A\subseteq A$, and associating
to a right ideal two-sided coideal $I$ the space of coinvariants associated to
the coalgebra surjection $A\to A/I$.

Let us now specialise to quantum homogeneous spaces. In this case the $B^{+}A$
will be a two-sided ideal and two-sided coideal. In fact, by Koppinen’s lemma
(see [31, Lemma 1.4]) it will be a Hopf ideal, meaning that the quotient $A/I$
will be a Hopf algebra. To go in the other direction, we need the following
technical lemma.

######  Lemma A.1.

For any surjective Hopf map $\pi:A\to H$, the associated space of coinvariants
$B=A^{\mathrm{co}(H)}$ satisfies $B^{+}A=AB^{+}$.

###### Proof.

For any $a\in A$ and $b\in B^{+}$, it holds that
$ab=a_{(1)}bS(a_{(2)})a_{(3)},$ and that $\varepsilon(a_{(1)}bS(a_{(2)}))=0$.
Moreover, we see that

$\displaystyle\Delta_{R,H}(a_{(1)}bS(a_{(2)}))=$
$\displaystyle\,a_{(1)}b_{(1)}S(a_{(4)})\otimes\pi(a_{(2)})\pi(b_{(2)})\pi(S(a_{(3)}))$
$\displaystyle=$
$\displaystyle\,a_{(1)}bS(a_{(4)})\otimes\pi(a_{(2)})\pi(S(a_{(3)}))$
$\displaystyle=$ $\displaystyle\,a_{(1)}bS(a_{(2)})\otimes 1.$

From this we can conclude that $ab=a_{(1)}bS(a_{(2)})a_{(3)}\in B^{+}A,$
implying the inclusion $AB^{+}\subseteq B^{+}A$. The opposite inclusion is
established analogously, giving the required equality. ∎

Thus we have a bijection between the quantum homogeneous $A$-spaces and the
set

$\Big{\\{}I\subseteq A\leavevmode\nobreak\ |\leavevmode\nobreak\ I\textrm{ a
Hopf ideal such that $A$ is faithfully coflat as a right
$A/I$-comodule}\Big{\\}}\\!.$

We finish with a second equivalent characterisation of quantum homogeneous
spaces. For any Hopf algebra $A$, a _quantum $A$-subgroup_ is a pair $(H,\pi)$
consisting of a Hopf algebra $H$ and and a surjective Hopf algebra map
$\pi:A\to H$, such that $A$ is faithfully coflat as a right
$A/\ker(\pi)$-comodule. Define an equivalence relation on quantum
$A$-subgroups by setting $(H,\pi)$ equivalent to $(H^{\prime},\pi^{\prime})$
if $\ker(\pi)=\ker(\pi^{\prime})$. Note that two quantum subgroups $(H,\pi)$
and $(H^{\prime},\pi^{\prime})$ are equivalent if and only if there exists a
Hopf algebra isomorphism $\phi:H\to H^{\prime}$ such that
$\phi\circ\pi=\pi^{\prime}$. This gives a bijective correspondence between
quantum homogeneous spaces and the set

$\Big{\\{}[(H,\pi)]\leavevmode\nobreak\ |\leavevmode\nobreak\ (H,\pi)\textrm{
is a quantum $A$-subgroup}\Big{\\}},$

where $[(H,\pi)]$ denotes the equivalence class of $(H,\pi)$.

###  A.3. Simple Objects of the Category ${}^{A}_{B}\mathrm{Mod}_{0}$

Since relative line modules are a primary object of interest in this paper, we
find it appropriate to make some general comments about simple objects in the
category ${}^{A}_{B}\mathrm{Mod}_{0}$, where as usual $A$ is a Hopf algebra,
and $B$ is a quantum homogeneous $A$-space

Note first that since $ab=a_{(1)}bS(a_{(2)})a_{(3)}$, for all $a\in A$, $b\in
B$, the Hopf algebra $A$ (endowed with its obvious right $B$-module structure)
is an object in ${}^{A}_{B}\mathrm{Mod}_{0}$. Moreover, any coideal
sub-$B$-bimodule of $A$ will also be an object in
${}^{A}_{B}\mathrm{Mod}_{0}$.

Conversely, consider a simple object
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}_{0}$. Since $\Phi({\mathcal{F}})$ is a
simple $\pi_{P}(A)$-comodule, it admits an embedding into $\pi_{P}(A)$,
implying that ${\mathcal{F}}$ embeds into $A$. Thus we see that any simple
object in ${}^{A}_{B}\mathrm{Mod}_{0}$ can be embedded as a sub-object of $A$.
In particular, all relative line modules embed into $A$. Note that, with
respect to an embedding ${\mathcal{F}}\hookrightarrow A$, it holds that
${\mathcal{F}}\simeq A\square_{\pi_{B}}\pi_{B}({\mathcal{F}}).$

Consider now the subcategory
${}^{A}_{B}\mathrm{mod}_{0}\subseteq{}^{A}_{B}\mathrm{Mod}_{0}$ of finite-
dimensional left $H$-comodules, and note that it is a rigid monoidal category.
From the monoidal form of Takeuchi’s equivalence, we see that the
corresponding subcategory
${}^{A}_{B}\mathrm{mod}_{0}\subseteq{}^{A}_{B}\mathrm{Mod}_{0}$ is also a
rigid monoidal category. Thus every
${\mathcal{F}}\in{}^{A}_{B}\mathrm{mod}_{0}$ admits a left and a right dual,
implying that ${\mathcal{F}}$ is finitely generated and projective as a left
and as a right $B$-module, and that ${}^{A}_{B}\mathrm{mod}_{0}$ can be
described as the subcategory whose objects are finitely generated left
$B$-modules. Moreover, any object in ${}^{A}_{B}\mathrm{Mod}_{0}$ which is a
direct sum of objects in ${}^{A}_{B}\mathrm{mod}_{0}$ will also be projective
as a left and as a right $B$-module.

Finally, we consider the case where $\pi_{B}(A)$ is a cosemisimple Hopf
algebra. The Peter–Weyl decomposition
$\pi_{B}(A)\simeq\bigoplus_{\alpha\in\widehat{\pi_{B}}}\pi_{B}(A)_{\alpha}$
gives a corresponding decomposition of $A$ into simple sub-objects

$\displaystyle A\simeq\bigoplus_{\alpha\in\widehat{\pi_{B}}}A_{\alpha},$

where we have denoted

$A_{\alpha}:=\left\\{a\in
A\,|\,\pi_{B}(a)\in\pi_{B}(A)_{\alpha}\right\\}=\\{a\in
A\,|\,a_{(1)}\otimes\pi_{B}(a_{(2)})\in A\otimes\pi_{B}(A)_{\alpha}\\}.$

It is instructive to note that

$\Phi(A_{\alpha})=\pi_{B}(A_{\alpha})=\pi_{B}(A)_{\alpha}.$

Cosemisimlicity of $\pi_{B}(A)$ implies that $A$ is isomorphic to the direct
sum of the simple objects in ${}^{A}_{B}\mathrm{mod}_{0}$. Moreover, since
each simple object is projective as a left and as a right $B$-module, each
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}_{0}$ must be projective as a left and
as a right $B$-module. Thus we have recovered some well-known results about
relative Hopf modules (see for example [15, Corollary 2.4] and [31, Corollary
1.5]) from the monoidal form of Takeuchi’s equivalence.

###  A.4. Connections and Projectivity

Principal $\ell$-maps are the link between principal comodule algebras and
noncommutative geometry. In particular, principal $\ell$-maps are in bijective
correspondence with principal connections
$\Pi:\Omega^{1}_{u}(P)\to\Omega^{1}_{u}(P)$, where

$\Omega^{1}_{u}(P)=\ker(m:P\otimes P\to P)$

is the universal first-order differential calculus over $P$. Principal
connections in turn provide a systematic procedure for constructing
connections
$\nabla:{\mathcal{F}}\to\Omega^{1}_{u}(P)\otimes_{P}{\mathcal{F}}$, where
${\mathcal{F}}$ is an _associated module_ , which is to say a module of the
form $P\square_{H}V_{{\mathcal{F}}}$, for some left $H$-comodule
$V_{{\mathcal{F}}}$. See the recent monograph [3, Chapter 5], or the
associated papers [12, 13], for more details.

It now follows from the Cuntz–Quillen theorem [18, §8.3] that any associated
module is projective as a left $B$-module. In particular, since any quantum
homogeneous space is a principal comodule algebra, every relative Hopf module
is projective as a left $B$-module.

## References

  * [1] F. Arici, S. Brain, and G. Landi, The Gysin sequence for quantum lens spaces, J. Noncommut. Geom., 9 (2015), pp. 1077–1111.
  * [2] F. Arici and J. Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, 2020.  arXiv preprint math.OA/2012.11186, 2020.
  * [3] E. Beggs and S. Majid, Quantum Riemannian Geometry, vol. 355 of Grundlehren der mathematischen Wissenschaften, Springer International Publishing, 1 ed., 2019.
  * [4] G. Böhm and T. Brzeziński, Strong connections and the relative Chern–Galois character for corings, Int. Math. Res. Not., (2005), pp. 2579–2625.
  * [5] T. Brzeziński, On modules associated to coalgebra Galois extensions, J. Algebra, 215 (1999), pp. 290–317.
  * [6] T. Brzeziński and P. M. Hajac, The Chern–Galois character, C. R. Math. Acad. Sci. Paris, 338 (2004), pp. 113–116.
  * [7] T. Brzeziński, G. Janelidze, and T. Maszczyk, Galois structures, in Lecture notes on noncommutative geometry and quantum groups, P. M. Hajac, ed., 2008.  Available at https://www.mimuw.edu.pl/~pwit/toknotes/toknotes.pdf.
  * [8] T. Brzeziński and S. Majid, Quantum group gauge theory on quantum spaces, Communications in Mathematical Physics, 157 (1993), pp. 591–638.
  * [9] , Quantum differentials and the $q$-monopole revisited, Acta Appl. Math., 54 (1998), pp. 185–232.
  * [10] T. Brzeziński and W. Szymański, On the quantum flag manifold ${SU_{q}(3)/\mathbb{T}^{2}}$, in Geometric methods in physics XXXVII, Trends Math., Birkhäuser/Springer, Cham, 2019, pp. 129–139.
  * [11] , An algebraic framework for noncommutative bundles with homogeneous fibres, Algebra Number Theory, 15 (2021), pp. 217–240.
  * [12] A. Carotenuto, F. Díaz García, and R. Ó Buachalla, A Borel–Weil theorem for the irreducible quantum flag manifolds.  In preparation.
  * [13] A. Carotenuto, C. Mrozinski, and R. Ó Buachalla, A Borel–Weil theorem for the quantum Grassmannians.  arXiv preprint math.QA/1611.07969.
  * [14] S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969\.
  * [15] L. Da̧browski, H. Grosse, and P. M. Hajac, Strong connections and Chern–Connes pairing in the Hopf–Galois theory, Comm. Math. Phys., 220 (2001), pp. 301–331.
  * [16] F. Díaz García, A. Krutov, R. Ó Buachalla, P. Somberg, and K. R. Strung, Holomorphic relative Hopf modules over the irreducible quantum flag manifolds, Lett. Math. Phys., 111 (2021), pp. Paper No. 10, 24.
  * [17] M. Dijkhuizen and J. Stokman, Quantized flag manifolds and irreducible $*$-representations, Comm. Math. Phys. (2), 203 (1999), pp. 297–324.
  * [18] H. Figueroa, J. M. Gracia-Bondía, and J. C. Várilly, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2001.
  * [19] V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Foundations of Lie theory and Lie transformation groups, Springer-Verlag, Berlin, 1997\.  Translated from the Russian by A. Kozlowski, Reprint of the 1993 translation [ıt Lie groups and Lie algebras. I, Encyclopaedia Math. Sci., 20, Springer, Berlin, 1993].
  * [20] P. M. Hajac and S. Majid, Projective module description of the $q$-monopole, Comm. Math. Phys., 206 (1999), pp. 247–264.
  * [21] I. Heckenberger and S. Kolb, The locally finite part of the dual coalgebra of quantized irreducible flag manifolds, Proc. London Math. Soc. (3), 89 (2004), pp. 457–484.
  * [22] , De Rham complex for quantized irreducible flag manifolds, J. Algebra, 305 (2006), pp. 704–741.
  * [23] J. E. Humphreys, Introduction to Lie algebras and representation theory, vol. 9 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1978.  Second printing, revised.
  * [24] A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Texts and Monographs in Physics, Springer-Verlag, 1997.
  * [25] H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J., 30 (1981), pp. 675–692.
  * [26] G. Landi, C. Pagani, and C. Reina, A Hopf bundle over a quantum four-sphere from the symplectic group, Comm. Math. Phys., 263 (2006), pp. 65–88.
  * [27] S. Majid, Noncommutative Riemannian and spin geometry of the standard $q$-sphere, Comm. Math. Phys., 256 (2005), pp. 255–285.
  * [28] A. Masuoka, Quotient theory of Hopf algebras, in Advances in Hopf algebras (Chicago, IL, 1992), vol. 158 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1994, pp. 107–133.
  * [29] S. E. Mikkelsen and W. Szymański, The quantum twistor bundle.  arXiv preprint math.QA/2005.03268, 2021.
  * [30] S. Montgomery, Hopf algebras and their actions on rings, vol. 82 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993.
  * [31] E. F. Müller and H.-J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math., 111 (1999), pp. 157–190.
  * [32] R. Ó Buachalla, Quantum bundle description of quantum projective spaces, Comm. Math. Phys., 316 (2012), pp. 345–373.
  * [33] , Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys., 99 (2016), pp. 154–173.
  * [34] R. Ó Buachalla, J. Šťoviček, and A.-C. van Roosmalen, A Kodaira vanishing theorem for noncommutative Kähler structures.  arXiv preprint math.QA/1801.08125, 2018.
  * [35] G. B. Podkolzin and L. I. Vainerman, Quantum Stiefel manifold and double cosets of quantum unitary group, Pacific J. Math., 188 (1999), pp. 179–199.
  * [36] B. Saurabh, Topological invariance of quantum quaternion spheres, Pacific J. Math., 288 (2017), pp. 435–452.
  * [37] , Quantum Stiefel manifolds, J. Ramanujan Math. Soc., 35 (2020), pp. 201–216.
  * [38] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math., vol. 72, 1990, pp. 167–195.  Hopf algebras.
  * [39] J. V. Stokman, The quantum orbit method for generalized flag manifolds, Math. Res. Lett., 10 (2003), pp. 469–481.
  * [40] M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math., 7 (1972), pp. 251–270.
  * [41] M. Takeuchi, Relative Hopf modules–equivalences and freeness criteria, J. Algebra, 60 (1979), pp. 452–471.
  * [42] L. L. Vaksman and Y. S. Soĭbelman, Algebra of functions on the quantum group ${\rm SU}(n+1),$ and odd-dimensional quantum spheres, Algebra i Analiz, 2 (1990), pp. 101–120.