Differential Calculi on Quantum Principal Bundles

over Projective Bases

P. Aschieri$,^{\\!1,2,3}$ R. Fioresi$,^{\\!4}$ E. Latini$,^{4}$ T. Weber1,2

1Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte
Orientale

Viale T. Michel - 15121, Alessandria, Italy

<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1,
10125 Torino

3Arnold-Regge Center, Torino, via P. Giuria 1, 10125, Torino, Italy

4Dipartimento di Matematica, Università di Bologna

Piazza di Porta S. Donato 5, I-40126 Bologna, Italy

<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We propose a sheaf-theoretic approach to the theory of differential calculi on
quantum principal bundles over non-affine bases. Our main class of examples is
given by quantum principal bundles over quantum flags. Given a first order
differential calculus on the total space of the bundle, covariant under the
action of the quantum structure group, if the base is affine we construct
calculi on the base and on the structure Hopf algebra. If the base is
projective we construct calculi on the sheaf of localizations, these induce
canonical differential structures on the base sheaf and on the structure Hopf
algebra and vice versa. The example of the quantum principal bundle
$\mathcal{O}_{q}(\mathrm{SL}_{2})$ over $\mathrm{P}^{1}(\mathbb{C})$ is
discussed in detail.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary concepts
    1. 2.1 Notation on modules and comodules
    2. 2.2 Covariant bimodules
    3. 2.3 Hopf-Galois extension
    4. 2.4 Smash product
  3. 3 Covariant differential calculi on Hopf-Galois extensions
    1. 3.1 Noncommutative differential calculi
    2. 3.2 Covariant calculi on comodule algebras
    3. 3.3 Bicovariant calculi
    4. 3.4 The smash product calculus
  4. 4 A sheaf-theoretic approach to noncommutative calculi
    1. 4.1 Differential calculus on sheaves
    2. 4.2 Ore extension of covariant calculi
    3. 4.3 Quantum differential calculi on QPB over projective varieties
    4. 4.4 Example of a differential structure on sheaves

##  1 Introduction

In this paper we introduce noncommutative differential calculi on quantum
principal bundles first in the affine setting, that is with affine base, along
the lines and extending the works [55, 29, 47]; then we employ the sheaf-
theoretic language developed in [3] to extend our definition and results to
non-affine bases. In noncommutative geometry, in fact, affine geometric
objects are replaced by the algebra of functions on them, algebraic or
differential. However, when it is necessary to go beyond the affine setting,
we must take the ringed space approach as expressed by Grothendieck: to do
geometry one does not need the space itself, but only the category of sheaves
on that would-be space, (see [25] and also [52] with refs. therein for more
insight on this point of view).

Important steps towards a sound foundation of noncommutative algebraic
geometry were taken in the works [1, 35, 49, 53]. In [1], quantum projective
schemes are understood as the category of modules associated with a given
graded noncommutative algebra, while in [49], an affine quantum space is
viewed as the spectrum of a noncommutative ring and quantum projective schemes
are defined accordingly with a gluing procedure in analogy with the classical
$\mathrm{Proj}$ construction. Indeed, this is our perspective on non-affine
noncommutative spaces: we shall define, following [3], a quantum space as a
locally ringed space and then build quantum principal bundles over quantum
spaces as sheaves with suitable (co)invariant properties. In [46] Pflaum
discusses a sheaf-theoretic approach on noncommutative geometry and in
particular gives a characterization of principal bundles, which is expecially
suited for the noncommutative setting and was exploited in [3] to go beyond
the case of affine base. In fact, though we have extensive treatments on
quantum principal bundles [7, 6, 8, 27] in the affine setting, the more
general, i.e. non-affine case, lacks a comprehensive and exhaustive treatment
(see also [26]). In [3] we study this more general setting, in particular
examining the important example of quantum flags of algebraic groups (see also
[49, 18, 20]) with total (classical) space the group itself. In such a setting
we show that the quantum space has an open cover with opens trivializing the
classical underlying principal bundle. A graded quantum ring is then
constructed out of the key notion of quantum section, associated to a quantum
line bundle (see [13]), with a gluing procedure similar to the classical
$\mathrm{Proj}$ one in [49].

In the present paper we exploit such quantum section construction in order to
obtain an important class of examples of the sheaf-theoretic approach to
differential calculi on quantum principal bundles that we propose. We start
with the discussion of first order differential calculus (FODC) in the affine
setting, that is, when our base and total spaces are affine. This case has
been fruitfully investigated in a series of works including [14, 33, 37, 43,
47] therefore this is an introductory section in which we mainly summarize all
the relevant results and establish our notation (see [2]).

Then, we make use of these affine case results and proceed to introduce and
study the sheaf approach to FODC on quantum principal bundles. The main
example follows: a quantum differential calculus on quantum flags for quantum
algebraic groups (see also [29, 30, 31, 44] for such example from a different
point of view).

We summarize the affine case results of Section 3 and 4 in the following
theorem.

######  Theorem 1.1.

Let $H$ be a Hopf algebra, $A$ a right $H$-comodule algebra and
$B:=A^{\mathrm{co}H}\subseteq A$ a trivial Hopf–Galois extension.

  1. i.)

Any right $H$-covariant FODC $(\Gamma,\mathrm{d})$ on $A$ induces a FODC
$(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and a right $H$-covariant FODC
$(\Gamma_{H},\mathrm{d}_{H})$ on $H$.

  2. ii.)

Vice-versa, a FODC $(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and a right
$H$-covariant FODC $(\Gamma_{H},\mathrm{d}_{H})$ on $H$ induces a right
$H$-covariant FODC $(\Gamma,\mathrm{d})$ on $A$.

  3. iii.)

These procedures are inverse to each other.

The calculi in i.) are obtained from the algebra embeddings $\iota\colon
B\hookrightarrow A$ and $j\colon H\rightarrow A$, via a general construction
of pullback calculus. The result in ii.) is obtained constructing the smash
product calculus $(\Gamma_{\\#},\mathrm{d}_{\\#})$ from the differential
calculi on $B$ and $H$. The right $H$-comodule algebra isomorphism $A\cong
B\\#H$ leads to the third statement of Theorem 1.1.

In the last section, we present the sheaf-theoretic approach to FODC on
quantum principal bundles (QPB) along the vein of [3], where a QPB is
understood as a locally cleft sheaf $\mathcal{F}$ of $H$-comodule algebras on
a quantum ringed space $(M,\mathcal{O}_{M})$. We view a FODC on $\mathcal{F}$
as a sheaf $\Upsilon$ on the same quantum ringed space $(M,\mathcal{O}_{M})$
with a natural transformation
$\mathrm{d}\colon\mathcal{F}\longrightarrow\Upsilon$, satisfying the Leibniz
property on an open cover (see Definition 4.1).

Once we have defined FODC on QPB on a base, not necessarily affine, we then
show how in the affine case we recover the differential calculi of Sections 2
and 3.

The main result in this section regards the special case of the quantization
of the classical principal bundle $p\colon G\longrightarrow G/P$, where $G$ is
a simple complex algebraic group and $P$ a parabolic subgroup, so that $G/P$
is a projective variety. Let $\mathcal{O}_{q}(G)$ be a quantum deformation of
$G$ and $\mathcal{O}_{q}(P)=\mathcal{O}_{q}(G)/I_{P}$ a quantum deformation of
a parabolic subgroup; both $\mathcal{O}_{q}(G)$ and $\mathcal{O}_{q}(P)$ are
Hopf algebras (see [13] Sec. 3 and [3]). Via the datum of a quantum section
$s\in\mathcal{O}_{q}(G)$, we obtain a quantization of a very ample classical
line bundle associated to a given classical projective embedding
$G/P\hookrightarrow\mathbf{P}^{n}$. Through the quantum section in fact, we
define a graded algebra $\mathcal{O}_{q}(G/P)$, which is a quantization of the
graded algebra obtained from the projective embedding
$G/P\hookrightarrow\mathbf{P}^{n}$: each graded component consists of the
elements of $\mathcal{O}_{q}(G)$, which are not coinvariant, but transform
according to powers of the projection of $s$ on $\mathcal{O}_{q}(P)$.

Furthermore, the quantum section $s$ allows to define two sheaves
$\mathcal{F}_{G}$ and $\mathcal{O}_{M}$ on $M=G/P$ and in [3] we prove that
$\mathcal{F}_{G}$ is a sheaf of locally cleft $\mathcal{O}_{q}(P)$-comodule
algebras on the quantum ringed space $(M,\mathcal{O}_{M})$, i.e. a quantum
principal bundle. In the construction of the sheaf $\mathcal{F}_{G}$, we use
the quantum section $s$ to define an open cover $\\{V_{i}\\}$ of $G$ and the
projected one $\\{U_{i}:=p(V_{i})\\}$ of $G/P$ (where
$G\stackrel{{\scriptstyle p}}{{\longrightarrow}}G/P$); the intersections
$U_{I}:=U_{i_{1}}\cap\dots\cap U_{i_{r}}$, $I=(i_{1},\dots,i_{r})$ form the
base for the topology we use to define both the sheaves $\mathcal{F}_{G}$ and
$\mathcal{O}_{M}$.

The main result for this part is the following theorem.

######  Theorem 1.2.

Let $(\Gamma,d)$ be a right covariant FODC on the Hopf algebra
$\mathcal{O}_{q}(G)$ and $s$ a quantum section. Then

  1. i.)

there is a right $\mathcal{O}_{q}(P)$-covariant FODC
$(\Upsilon_{G},\mathrm{d}_{G})$ on the sheaf $\mathcal{F}_{G}$, where the
bimodule $\Upsilon_{G}(U_{I})$ is the extension of the bimodule $\Gamma$
defined by

$\Upsilon_{G}(U_{I}):=\mathcal{F}_{G}(U_{I})\,\Gamma\,\mathcal{F}_{G}(U_{I})$
(1)

and the exterior derivatives
$\mathrm{d}_{I}:=\mathrm{d}_{G}|_{U_{I}}\colon\mathcal{O}_{q}(U_{I})\rightarrow\Upsilon_{G}(U_{I})$,
$I=(i_{1},\ldots,i_{r})$, are extensions of
$\mathrm{d}\colon\mathcal{O}_{q}(G)\to\Gamma$;

  2. ii.)

the FODC $(\Upsilon_{G},\mathrm{d}_{G})$ on the sheaf $\mathcal{F}_{G}$
induces a FODC $(\Upsilon_{M},\mathrm{d}_{M})$ on the sheaf $\mathcal{O}_{M}$,
where

$\Upsilon_{M}(U_{I}):=\mathcal{O}_{M}(U_{I})\,\mathrm{d}_{I}\,\mathcal{O}_{M}(U_{I})$
(2)

and
$(\mathrm{d}_{M})_{I}:=\mathrm{d}_{I}|_{\mathcal{O}_{M}(U_{I})}\colon\mathcal{O}_{M}(U_{I})\rightarrow\Upsilon_{M}(U_{I})$,
$I=(i_{1},\ldots,i_{r})$;

The results of Theorem 1.1 hold true also in this more general setting.

######  Theorem 1.3.

There is a one to one correspondence between:

  1. 1.

A right $\mathcal{O}_{q}(P)$-covariant FODC $(\Upsilon,\mathrm{d})$ on the
sheaf $\mathcal{F}_{G}$.

  2. 2.

A FODC $(\Upsilon_{M},\mathrm{d}_{M})$ on the sheaf $\mathcal{O}_{M}$ and a
right $\mathcal{O}_{q}(P)$-covariant FODC $(\Gamma_{P},\mathrm{d}_{P})$ on
$\mathcal{O}_{q}(P)$.

Note that we obtain a constant sheaf $(\Gamma_{P},\mathrm{d}_{P})$ on the
structure Hopf algebra $\mathcal{O}_{q}(P)$ since the calculi induced from
$\Upsilon_{G}(U_{I})$ on $\mathcal{O}_{q}(P)$ are all isomorphic.

##  2 Preliminary concepts

In this section we establish our notation and provide the key results we will
need in the sequel. Though most of the material of this section is known (see
[40, 41]), we include it by completeness, so that the reader has readily
available the results in the form we need them later on.

###  2.1 Notation on modules and comodules

In this paper $\Bbbk$ denotes a commutative unital ring and $\otimes$ is the
tensor product of $\Bbbk$-modules. Let $H$ be a Hopf algebra over $\Bbbk$.
Throughout this paper we denote the coproduct of $H$ by $\Delta$ and its
counit by $\epsilon$. Recall that those are algebra homomorphisms
$\Delta\colon H\rightarrow H\otimes H$ and $\epsilon\colon H\rightarrow\Bbbk$
such that
$(\Delta\otimes\mathrm{id}_{H})\circ\Delta=(\mathrm{id}_{H}\otimes\Delta)\circ\Delta$
(coassociativity) and
$(\epsilon\otimes\mathrm{id}_{H})\circ\Delta=(\mathrm{id}_{H}\otimes\epsilon)\circ\Delta=\mathrm{id}_{H}$
(counitality) hold. We employ Sweedler’s notation $\Delta(a)=a_{1}\otimes
a_{2}$ for the coproduct of an element $a\in H$, i.e. we omit summation
symbols and summation indices. By the coassociativity we write

$a_{1}\otimes a_{2}\otimes
a_{3}:=(\Delta\otimes\mathrm{id}_{H})(\Delta(a))=(\mathrm{id}_{H}\otimes\Delta)(\Delta(a))$
(3)

for all $a\in H$ and similarly for higher coproducts. The antipode $S\colon
H\rightarrow H$ of $H$ is an anti-bialgebra homomorphism, namely
$S(ab)=S(b)S(a)$, $S(1)=1$, $\Delta(S(a))=S(a_{2})\otimes S(a_{1})$,
$\epsilon\circ S=\epsilon$, and furthermore we have
$S(a_{1})a_{2}=\epsilon(a)1=a_{1}S(a_{2})$ for all $a,b\in H$. If $S$ is
invertible as a $\Bbbk$-linear map, its inverse $\overline{S}\colon
H\rightarrow H$ is an anti-bialgebra homomorphism such that
$\overline{S}(a_{2})a_{1}=\epsilon(a)1=a_{2}\overline{S}(a_{1})$ for all $a\in
H$.

A right $H$-comodule is a $\Bbbk$-module $M$, together with a $\Bbbk$-linear
map $\Delta_{R}\colon M\rightarrow M\otimes H$ such that

$(\Delta_{R}\otimes\mathrm{id}_{H})\circ\Delta_{R}=(\mathrm{id}_{M}\otimes\Delta)\circ\Delta_{R}$
(4)

and $(\mathrm{id}_{M}\otimes\epsilon)\circ\Delta_{R}=\mathrm{id}_{M}$, while a
left $H$-comodule is a $\Bbbk$-module $M$ with a $\Bbbk$-linear map
$\Delta_{L}\colon M\rightarrow H\otimes M$ such that

$(\mathrm{id}_{H}\otimes\Delta_{L})\circ\Delta_{L}=(\Delta\otimes\mathrm{id}_{M})\circ\Delta_{L}$
(5)

and $(\epsilon\otimes\mathrm{id}_{M})\circ\Delta_{L}=\mathrm{id}_{M}$. The
maps $\Delta_{R}$ and $\Delta_{L}$ are called right and left $H$-coaction,
respectively. We use the short notations $\Delta_{R}(m)=m_{0}\otimes m_{1}$
and $\Delta_{L}(m)=m_{-1}\otimes m_{0}$, where $m\in M$. In case $M$ is a
right, respectively left, $H$-comodule, we denote the $\Bbbk$-submodules of
right and left $H$-coinvariant elements by

$M^{\mathrm{co}H}:=\\{m\in M~{}|~{}\Delta_{R}(m)=m\otimes 1\\}$ (6)

and ${}^{\mathrm{co}H}M:=\\{m\in M~{}|~{}\Delta_{L}(m)=1\otimes m\\}$,
respectively. Following (4) and (5), we further abbreviate

$m_{0}\otimes m_{1}\otimes
m_{2}:=(\Delta_{R}\otimes\mathrm{id}_{H})(\Delta_{R}(m))=(\mathrm{id}_{M}\otimes\Delta)(\Delta_{R}(m)),$
(7)

$m_{-2}\otimes m_{-1}\otimes
m_{0}:=(\mathrm{id}_{H}\otimes\Delta_{L})(\Delta_{L}(m))=(\Delta\otimes\mathrm{id}_{M})(\Delta_{L}(m))$
and similarly for higher coactions. A right $H$-comodule $M$ which is also a
left $H$-comodule is said to be an $H$-bicomodule if $\Delta_{R}$ and
$\Delta_{L}$ are commuting coactions, i.e.

$(\Delta_{L}\otimes\mathrm{id}_{H})\circ\Delta_{R}=(\mathrm{id}_{H}\otimes\Delta_{R})\circ\Delta_{L}.$
(8)

In short notation, (8) reads $m_{-1}\otimes m_{0}\otimes
m_{1}:=(\Delta_{L}\otimes\mathrm{id}_{H})(\Delta_{R}(m))=(\mathrm{id}_{H}\otimes\Delta_{R})(\Delta_{L}(m))$.
A right $H$-comodule morphism is a $\Bbbk$-linear map $\phi\colon M\rightarrow
N$ between right $H$-comodules $(M,\Delta_{M})$ and $(N,\Delta_{N})$ such that

$\phi(m)_{0}\otimes\phi(m)_{1}=\phi(m_{0})\otimes m_{1}$ (9)

for all $m\in M$. Note that on the left hand side of (9) we used the short
notation $\Delta_{N}(\phi(m))=\phi(m)_{0}\otimes\phi(m)_{1}$, while on the
right hand side of (9) we used $\Delta_{M}(m)=m_{0}\otimes m_{1}$. A
$\Bbbk$-linear map satisfying (9) is sometimes called right $H$-colinear.
Similarly, left $H$-colinear and $H$-bicolinear maps are defined.

###  2.2 Covariant bimodules

Algebras in the category of comodules have a multiplication and unit that are
compatible with the coaction. Since we are encountering several different
coactions in the course of this section we utilize the convention to denote
the right $H$-coaction on a right $H$-comodule algebra $A$ by
$\delta_{R}\colon A\rightarrow A\otimes H$.

######  Definition 2.1.

A right $H$-comodule algebra is a right $H$-comodule $(A,\delta_{R})$ together
with an associative product $\mu\colon A\otimes A\rightarrow A$ and a unit
$\eta\colon\Bbbk\rightarrow A$, such that

$\delta_{R}\circ\mu=(\mu\otimes\mu_{H})\circ(\mathrm{id}_{A}\otimes\tau_{H,A}\otimes\mathrm{id}_{H})\circ(\delta_{R}\otimes\delta_{R})$
(10)

and $\delta_{R}\circ\eta=\eta\otimes\eta_{H}$, where $\tau_{H,A}\colon
H\otimes A\rightarrow A\otimes H$ denotes the flip isomorphism and
$(\mu_{H},\eta_{H})$ is the algebra structure of $H$. Using the previously
introduced short notation, the compatibility conditions read

$\delta_{R}(ab)=(ab)_{0}\otimes(ab)_{1}=a_{0}b_{0}\otimes
a_{1}b_{1}=\delta_{R}(a)\delta_{R}(b)$ (11)

for all $a,b\in A$ and $\delta_{R}(1)=1_{0}\otimes 1_{1}=1\otimes 1_{H}$. A
right $H$-comodule algebra homomorphism is a right $H$-comodule homomorphism
which is also an algebra homomorphism. Similarly, one defines left
$H$-comodule algebras and $H$-bicomodule algebras and their morphisms.

Fix a right $H$-comodule algebra $(A,\delta_{R})$ in the following. We now
introduce $A$-bimodules in the category of right $H$-comodules.

######  Definition 2.2.

An $A$-bimodule $M$ is called a right $H$-covariant $A$-bimodule if there is a
right $H$-comodule action $\Delta_{R}\colon M\rightarrow M\otimes H$ on $M$,
such that

$\Delta_{R}(a\cdot m\cdot b)=\delta_{R}(a)\Delta_{R}(m)\delta_{R}(b),$ (12)

or equivalently, in short notation, $(a\cdot m\cdot b)_{0}\otimes(a\cdot
m\cdot b)_{1}=a_{0}\cdot m_{0}\cdot b_{0}\otimes a_{1}m_{1}b_{1}$ for all
$a,b\in A$ and $m\in M$. A right $H$-covariant right (resp. left) $A$-module
is a right (resp. left) $A$-module, where we only ask compatibility of a right
$H$-coaction with the right (resp. left) $A$-module structure. Similarly, left
$H$-covariant $A$-modules and $H$-bicovariant $A$-bimodules are defined, if
$A$ is a left $H$-comodule algebra or an $H$-bicomodule algebra, respectively.

We are particularly interested in the case $A=H$, since $H$ is naturally a
right and a left $H$-comodule algebra with respect to the coactions given by
the coproduct.

######  Definition 2.3.

A bicovariant $H$-bimodule is an $H$-bimodule and an $H$-bicomodule $M$, such
that

$\Delta_{R}(a\cdot m\cdot b)=\Delta(a)\Delta_{R}(m)\Delta(b)\text{ and
}\Delta_{L}(a\cdot m\cdot b)=\Delta(a)\Delta_{L}(m)\Delta(b)$ (13)

for all $a,b\in H$ and $m\in M$.

For $H$-covariant $H$-modules, there is the following fundamental theorem
(c.f. [40] Theorem 1.9.4, [50]).

######  Proposition 2.4.

For any right $H$-covariant right $H$-module $M$, there is an isomorphism

$M\ni m\mapsto m_{0}S(m_{1})\otimes m_{2}\in M^{\mathrm{co}H}\otimes H,$ (14)

of right $H$-covariant right $H$-modules. If $M$ is a bicovariant
$H$-bimodule, (14) extends to an isomorphism of bicovariant $H$-bimodules.

If $M$ is a left $H$-covariant left $H$-module, there is an isomorphism

$M\ni m\mapsto m_{-2}\otimes S(m_{-1})m_{0}\in H\otimes{}^{\mathrm{co}H}M,$
(15)

of left $H$-covariant left $H$-modules. If $M$ is a bicovariant $H$-bimodule,
(15) extends to an isomorphism of bicovariant $H$-bimodules.

Note that in (14) the tensor product $M^{\mathrm{co}H}\otimes H$ is a right
$H$-covariant right $H$-module with respect to the diagonal coaction and the
right $H$-action $(\overline{m}\otimes h)\cdot
h^{\prime}:=\overline{m}\otimes(hh^{\prime})$, where $\overline{m}\in
M^{\mathrm{co}H}$ and $h,h^{\prime}\in H$. If $M$ is even a bicovariant
$H$-bimodule we endow $M^{\mathrm{co}H}\otimes H$ with the diagonal left
$H$-coaction and the left $H$-action $h\cdot(\overline{m}\otimes
h^{\prime}):=h_{1}\overline{m}S(h_{2})\otimes h_{3}h^{\prime}$. Similarly in
(15), $H\otimes{}^{\mathrm{co}H}M$ becomes a left $H$-covariant left
$H$-module via the diagonal coaction and the left $H$-action given by
$h\cdot(h^{\prime}\otimes\overline{m}):=(hh^{\prime})\otimes\overline{m}$,
where $h,h^{\prime}\in H$ and $\overline{m}\in{}^{\mathrm{co}H}M$. If $M$ is
even a bicovariant $H$-bimodule we endow $H\otimes{}^{\mathrm{co}H}M$ with the
diagonal right $H$-coaction and the right $H$-action
$(h\otimes\overline{m})\cdot h^{\prime}:=hh^{\prime}_{1}\otimes
S(h^{\prime}_{2})\overline{m}h^{\prime}_{3}$.

Using the inverse antipode $\overline{S}$ there are analogous isomorphism of
right $H$-covariant left $H$-modules and left $H$-covariant right $H$-modules,
respectively. The fundamental theorem allows us to write any element $m$ of a
bicovariant $H$-bimodule as a finite sum $m=a_{i}m^{i}$, where
$\\{m^{i}\\}_{i}\subseteq{}^{\mathrm{co}H}M$ is a basis of (left) coinvariant
elements and $a_{i}$ are coefficients in $H$. This is particularly useful when
one is dealing with $H$-linear maps of bicovariant bimodules. Then, it is
sufficient to specify properties on coinvariant elements.

###  2.3 Hopf-Galois extension

In this section $H$ denotes a Hopf algebra and $A$ a right $H$-comodule
algebra with right $H$-comodule action $\delta_{R}\colon A\rightarrow A\otimes
H$.

######  Definition 2.5.

The algebra extension $B:=A^{\mathrm{co}H}\subseteq A$ is said to be

  1. i.)

a Hopf-Galois extension, if the $\Bbbk$-linear map

$\chi:=(\mu_{A}\otimes\mathrm{id}_{H})\circ(\mathrm{id}_{A}\otimes_{B}\delta_{R})\colon
A\otimes_{B}A\rightarrow A\otimes H$ (16)

is a bijection, where $\mu_{A}\colon A\otimes A\rightarrow A$ denotes the
product of $A$;

  2. ii.)

a cleft extension, if there is a $\Bbbk$-linear map $j\colon H\rightarrow A$,
the cleaving map, such that

    1. 1.)

$j$ is right $H$-colinear, i.e. $j(h)_{0}\otimes j(h)_{1}=j(h_{1})\otimes
h_{2}$ for all $h\in H$;

    2. 2.)

$j$ is convolution invertible, i.e. there is a $\Bbbk$-linear map
$j^{-1}\colon H\rightarrow A$, such that
$j(h_{1})j^{-1}(h_{2})=\epsilon(h)1=j^{-1}(h_{1})j(h_{2})$ for all $h\in H$;

    3. 3.)

$j$ respects the units, i.e. $j(1_{H})=1$;

  3. iii.)

a trivial extension, if there is a right $H$-comodule algebra homomorphism
$j\colon H\rightarrow A$;

In short notation (16) reads
$\chi(a\otimes_{B}a^{\prime})=aa^{\prime}_{0}\otimes a^{\prime}_{1}$ for all
$a,a^{\prime}\in A$. A trivial extension $A^{\mathrm{co}H}\subseteq A$ is
automatically a cleft extension since an $H$-comodule algebra map $j\colon
H\to A$ is right $H$-colinear and unital by assumption and its convolution
inverse is $j^{-1}=j\circ S$. This also implies that
$j^{-1}\circ\overline{S}\colon H\rightarrow A$ is right $H$-colinear. In the
following lemma we prove that this is the case for any cleft extension.

######  Lemma 2.6.

The inverse of the cleaving map satisfies

$\delta_{R}\circ j^{-1}=(j^{-1}\otimes S)\circ\tau\circ\Delta.$ (17)

In particular, $j^{-1}\circ\overline{S}$ is right $H$-colinear.

###### Proof.

The equation (17) is proven in [40] Lemma 7.2.6, 1). Then,

$\displaystyle\delta_{R}(j^{-1}(\overline{S}(h)))=j^{-1}(\overline{S}(h)_{2})\otimes
S(\overline{S}(h)_{1})=j^{-1}(\overline{S}(h_{1}))\otimes
S(\overline{S}(h_{2}))=j^{-1}(\overline{S}(h_{1}))\otimes h_{2}$

for all $h\in H$, where we used that $\overline{S}$ is an anti-bialgebra
homomorphism. ∎

To complete the hierarchy of Definition 2.5, we note that cleft extensions are
in particular Hopf-Galois extensions.

######  Proposition 2.7.

$B:=A^{\mathrm{co}H}\subseteq A$ is a cleft extension if and only if
$B\subseteq A$ is a Hopf-Galois extension and there is a left $B$-module and
right $H$-comodule isomorphism $B\otimes H\cong A$.

The proof of the proposition (see e.g. [40, Theorem 8.2.4]) relies on the
construction of the following left $B$-module and right $H$-comodule map

$\theta\colon B\otimes H\to A\,,\,\,\,\,b\otimes h\mapsto bj(h)$ (18)

with inverse given by

$\theta^{-1}\colon A\to B\otimes H\,,\,\,\,\,a\mapsto
a_{0}j^{-1}(a_{1})\otimes a_{2}.$ (19)

Note that $\theta^{-1}$ is well-defined, i.e. $a_{0}j^{-1}(a_{1})\in B$, since

$\displaystyle\delta_{R}(a_{0}j^{-1}(a_{1}))=a_{0}j^{-1}(a_{2})_{0}\otimes
a_{1}j^{-1}(a_{2})_{1}=a_{0}j^{-1}(a_{3})\otimes
a_{1}S(a_{2})=a_{0}j^{-1}(a_{2})\otimes\epsilon(a_{1})1=a_{0}j^{-1}(a_{1})\otimes
1,$

where we employed Lemma 2.6 (see also [40] Lemma 7.2.6, 2.). The above
construction generalizes to right $H$-covariant $A$-modules.

######  Proposition 2.8.

Let $A$ be a right $H$-comodule algebra, $j\colon A\to H$ a cleaving map and
$M$ a right $H$-covariant right $A$-module, then

$M\cong M^{\mathrm{co}H}\otimes H$ (20)

is an isomorphism of right $H$-comodules, where $M^{\mathrm{co}H}\otimes H$ is
endowed with the diagonal coaction.

If $M$ is a right $H$-covariant $A$-bimodule, (20) becomes an isomorphism of
right $H$-comodules and left $B:=A^{\mathrm{co}H}$-modules, where
$M^{\mathrm{co}H}\otimes H$ is a left $B$-module via multiplication on the
first tensor factor.

###### Proof.

The isomorphism is given by $M\ni m\mapsto m_{0}j^{-1}(m_{1})\otimes m_{2}\in
M^{\mathrm{co}H}\otimes H$. In fact, $m_{0}j^{-1}(m_{1})\in M^{\mathrm{co}H}$,
since

$\displaystyle\Delta_{M}(m_{0}j^{-1}(m_{1}))=m_{0}j^{-1}(m_{2})_{1}\otimes
m_{1}j^{-1}(m_{2})_{2}=m_{0}j^{-1}(m_{3})\otimes
m_{1}S(m_{2})=m_{0}j^{-1}(m_{1})\otimes 1,$

where we used Lemma 2.6 and that $M$ is a right $H$-covariant right
$A$-module. The inverse isomorphism is given by $M^{\mathrm{co}H}\otimes
H\ni\overline{m}\otimes h\mapsto\overline{m}j(h)\in M$. Note that the left
$B$-module and right $H$-comodule structure on $M^{\mathrm{co}H}\otimes H$ is
given by

$b\cdot(\overline{m}\otimes h)=b\overline{m}\otimes h\text{ and
}(\overline{m}\otimes h)_{0}\otimes(\overline{m}\otimes
h)_{1}=\overline{m}\otimes h_{1}\otimes h_{2}$ (21)

for all $b\in B$, $h\in H$ and $\overline{m}\in M^{\mathrm{co}H}$. Then it is
obvious that the isomorphisms are left $B$-linear and right $H$-colinear,
which concludes the proof of the proposition. ∎

This can be understood as a generalization of the fundamental theorem of Hopf
modules (c.f. Proposition 2.4). Namely, even if an $H$-comodule only admits a
compatible $A$-module action, we obtain a splitting into coinvariant elements
and coefficient in $H$, if there is a cleaving map. The fundamental theorem of
Hopf modules is regained for $A=H$ and cleaving map $j=\mathrm{id}_{H}$ with
convolution inverse given by the antipode $S\colon H\rightarrow H$.

Note that (18) is left $B$-linear and right $H$-colinear, however not an
algebra isomorphism between $B\otimes H$ and $A$: in general, we have

$\theta((b\otimes h)(b^{\prime}\otimes h^{\prime}))=\theta(bb^{\prime}\otimes
hh^{\prime})=bb^{\prime}j(hh^{\prime})\neq
bj(h)b^{\prime}j(h^{\prime})=\theta(b\otimes h)\theta(b^{\prime}\otimes
h^{\prime})$

for $b,b^{\prime}\in B$ and $h,h^{\prime}\in H$. In the next section, we
modify the product on $B\otimes H$, in order to reconcile this problem in the
case of a trivial extension. This leads to the smash product. Note that there
is a crossed product construction on $B\otimes H$, which even reconciles the
general cleft case. However, we avoid this more involved approach since in
this paper, in particular in all examples, we encounter trivial extensions.

###  2.4 Smash product

Here we discuss the $H$-comodule algebra structure of the smash product
$B\\#H$, for more details on this well known construction see [40] Section
4.1. Fix a trivial extension $B:=A^{\mathrm{co}H}\subseteq A$ with cleaving
map $j\colon H\rightarrow A$ and convolution inverse $j^{-1}\colon
H\rightarrow A$. By assumption $j\colon H\rightarrow A$ is a right
$H$-colinear algebra homomorphism. However, note that in general $\theta\colon
B\otimes H\rightarrow A$ is not an algebra homomorphism, while the other
implication holds true: if $\theta$ is an algebra homomorphism, so is $j$,
since $j(hh^{\prime})=\theta(1\otimes hh^{\prime})=\theta(1\otimes
h)\theta(1\otimes h^{\prime})=j(h)j(h^{\prime})$ for all $h,h^{\prime}\in H$.
We define the $\Bbbk$-linear operation

$h\rhd b:=j(h_{1})bj^{-1}(h_{2})$ (22)

for all $h\in H$ and $b\in B$. Since $j$ is an algebra map and $j^{-1}=j\circ
S$ an anti-algebra map, it follows (see e.g. [40] Lemma 7.1.2 and Proposition
7.2.3) that (22) is a left $H$-action on $B$, where $j(h_{1})bj^{-1}(h_{2})\in
B$ by the right $H$-colinearity of $j$ and Lemma 2.6. Now we are prepared to
define the smash product algebra $B\\#H$.

######  Definition 2.9.

Given a trivial extension $B:=A^{\mathrm{co}H}\subseteq A$ we define the smash
product $B\\#H$ as the $\Bbbk$-module $B\otimes H$ endowed with the
multiplication

$(b\otimes h)\cdot_{\\#}(b^{\prime}\otimes h^{\prime}):=b(h_{1}\rhd
b^{\prime})\otimes h_{2}h^{\prime}$ (23)

for all $b,b^{\prime}\in B$ and $h,h^{\prime}\in H$.

In the following we write $b\\#h$ instead of $b\otimes h$ if we want to view
$b\otimes h$ as an element of $B\\#H$. Moreover, we set
$(b\\#h)(b^{\prime}\\#h^{\prime}):=(b\otimes h)\cdot_{\\#}(b^{\prime}\otimes
h^{\prime})$. We notice that $B\\#H$ is naturally endowed with the right
$H$-comodule structure

$B\\#H\ni b\\#h\mapsto(b\\#h_{1})\otimes h_{2}\in B\\#H\otimes H.$ (24)

This coaction is compatible with the product $\cdot_{\\#}$ and thus $B\\#H$ is
a right $H$-comodule algebra.

######  Proposition 2.10.

If $B:=A^{\mathrm{co}H}\subseteq A$ is a trivial extension there is an
isomorphism $\theta_{\\#}\colon B\\#H\rightarrow A$ of right $H$-comodule
algebras, given by $\theta_{\\#}(b\\#h):=bj(h)$ with inverse
$\theta^{-1}_{\\#}(a)=a_{0}j^{-1}(a_{1})\\#a_{2}$.

###### Proof.

In Proposition 2.7 it has been proven that $\theta_{\\#}$ is an isomorphism of
right $H$-comodules. It remains to prove that $\theta_{\\#}$ is an algebra
homomorphism. Clearly $\theta_{\\#}(1\\#1)=1$ and also

$\displaystyle\theta_{\\#}((b\\#h)(b^{\prime}\\#h^{\prime}))=$
$\displaystyle\theta_{\\#}(b(h_{1}\rhd
b^{\prime})\\#h_{2}h^{\prime})=b(h_{1}\rhd
b^{\prime})j(h_{2}h^{\prime})=bj(h_{1})b^{\prime}j^{-1}(h_{2})j(h_{3})j(h^{\prime})$
$\displaystyle=$ $\displaystyle
bj(h)b^{\prime}j(h^{\prime})=\theta_{\\#}(b\\#h)\theta_{\\#}(b^{\prime}\\#h^{\prime})$

holds. A full proof can be found in [40] Proposition 7.2.3. ∎

##  3 Covariant differential calculi on Hopf-Galois extensions

In this section we give our main results in the affine setting, namely
Proposition 3.13. We start by defining the category of noncommutative
differential calculi. In particular we discuss the construction of the
pullback calculus and the quotient calculus and give instances where both
constructions are isomorphic. Afterwards we specialize to covariant
differential calculi, generalizing the results in [29]. In case $A$ is a
trivial Hopf-Galois extension of $B:=A^{\mathrm{co}H}$, we are able to
establish a correspondence between FODC on $A$ and on $B$ and $H$. We then
proceed to a characterization of the base forms that is FODC on the base $B$.
Unless otherwise specified $A,A^{\prime},\ldots$ denote associative unital
algebras over a commutative unital ring $\Bbbk$ and $H$ is assumed to be a
$\Bbbk$-Hopf algebra.

###  3.1 Noncommutative differential calculi

In this section we give the definition and some results on noncommutative
differential calculi. Though this material is well established in the context
of bicovariant calculi on Hopf algebras, since we take a slightly more general
point of view, we prefer to recap the main points. We refer the reader to [4]
Chapter 2 for more details.

We give the definition of a differential calculus on a noncommutative algebra.

######  Definition 3.1.

A first order differential calculus (FODC) on $A$ is a tuple
$(\Gamma,\mathrm{d})$, where

  1. i.)

$\Gamma$ is an $A$-bimodule;

  2. ii.)

$\mathrm{d}\colon A\rightarrow\Gamma$ is a $\Bbbk$-linear map which satisfies
the Leibniz rule $\mathrm{d}(ab)=(\mathrm{d}a)b+a\mathrm{d}b$ for all $a,b\in
A$;

  3. iii.)

$\Gamma=A\mathrm{d}A:=\mathrm{span}_{\Bbbk}\\{a\mathrm{d}b~{}|~{}a,b\in A\\}$.
We say $\Gamma$ is generated by $A$ and $\mathrm{d}$;

A morphism between a FODC $(\Gamma,\mathrm{d})$ on $A$ and a FODC
$(\Gamma^{\prime},\mathrm{d}^{\prime})$ on another associative unital algebra
$A^{\prime}$ is a tuple $(\phi,\Phi)$, where $\phi\colon A\rightarrow
A^{\prime}$ is an algebra homomorphism and
$\Phi\colon\Gamma\rightarrow\Gamma^{\prime}$ is a $\Bbbk$-linear map such that

$\Phi\circ\mathrm{d}=\mathrm{d}^{\prime}\circ\phi$ (25)

and

$\Phi(a\cdot\omega\cdot b)=\phi(a)\cdot^{\prime}\omega\cdot^{\prime}\phi(b)$
(26)

for all $a,b\in A$. Two FODC are called equivalent if there is an isomorphism
of FODC between them.

The classical example of a FODC comes from differential geometry. Given a
smooth manifold $M$, the $C^{\infty}(M)$-bimodule $\Omega^{1}(M)$ of
differential $1$-forms on $M$, together with the de Rham differential, is a
FODC on $C^{\infty}(M)$. On any associative unital algebra $A$ there is the
so-called universal FODC $(\Gamma_{u},\mathrm{d}_{u})$, where
$\Gamma_{u}:=\ker\mu_{A}\subseteq A\otimes A$ is the kernel of the
multiplication $\mu_{A}\colon A\otimes A\rightarrow A$ and
$\mathrm{d}_{u}(a):=1\otimes a-a\otimes 1$ for all $a\in A$. The left and
right $A$-module action on $\Gamma_{u}$ is given by the multiplication on the
first, respectively and the second tensor factor. This FODC is universal in
the sense that every FODC on $A$ is isomorphic to a quotient of
$(\Gamma_{u},\mathrm{d}_{u})$ (c.f. [56]).

It is well-known, that the tensor product $A\otimes A^{\prime}$ of two
associative unital algebras $A$ and $A^{\prime}$ is an algebra with
associative product $(a\otimes a^{\prime})(b\otimes b^{\prime})=ab\otimes
a^{\prime}b^{\prime}$ and unit $1_{A}\otimes 1_{A^{\prime}}$. This
construction extends to the level of FODC (see also [47] Theorem 2.2).

######  Proposition 3.2.

Given a FODC $(\Gamma,\mathrm{d})$ on $A$ and a FODC
$(\Gamma^{\prime},\mathrm{d}^{\prime})$ on $A^{\prime}$, there is a FODC
$(\Gamma_{A\otimes A^{\prime}},\mathrm{d}_{A\otimes A^{\prime}})$ on $A\otimes
A^{\prime}$, where $\Gamma_{A\otimes A^{\prime}}=\Gamma\otimes
A^{\prime}\oplus A\otimes\Gamma^{\prime}$ and

$\mathrm{d}_{A\otimes A^{\prime}}\colon A\otimes A^{\prime}\ni a\otimes
a^{\prime}\mapsto\mathrm{d}a\otimes
a^{\prime}+a\otimes\mathrm{d}^{\prime}a^{\prime}\in\Gamma_{A\otimes
A^{\prime}}.$ (27)

The $A\otimes A^{\prime}$-bimodule structure on $\Gamma_{A\otimes A^{\prime}}$
is

$(a\otimes a^{\prime})\cdot(\omega\otimes
b^{\prime}+b\otimes\omega^{\prime})\cdot(c\otimes c^{\prime})=a\omega c\otimes
a^{\prime}b^{\prime}c^{\prime}+abc\otimes
a^{\prime}\omega^{\prime}c^{\prime},$ (28)

where $a,b,c\in A$, $a^{\prime},b^{\prime},c^{\prime}\in A^{\prime}$,
$\omega\in\Gamma$ and $\omega^{\prime}\in\Gamma^{\prime}$. This construction
is associative, i.e. $(\Gamma_{(A\otimes A^{\prime})\otimes
A^{\prime\prime}},\mathrm{d}_{(A\otimes A^{\prime})\otimes
A^{\prime\prime}})=(\Gamma_{A\otimes(A^{\prime}\otimes
A^{\prime\prime})},\mathrm{d}_{A\otimes(A^{\prime}\otimes A^{\prime\prime})})$
for another associative unital algebra $A^{\prime\prime}$.

###### Proof.

Clearly, (28) is an $A\otimes A^{\prime}$-bimodule structure and (27) is a
$\Bbbk$-linear map. The latter satisfies the Leibniz rule

$\displaystyle\mathrm{d}_{A\otimes A^{\prime}}((a\otimes a^{\prime})(b\otimes
b^{\prime}))=$ $\displaystyle\mathrm{d}_{A\otimes A^{\prime}}(ab\otimes
a^{\prime}b^{\prime})$ $\displaystyle=$ $\displaystyle\mathrm{d}(ab)\otimes
a^{\prime}b^{\prime}+ab\otimes\mathrm{d}^{\prime}(a^{\prime}b^{\prime})$
$\displaystyle=$ $\displaystyle(\mathrm{d}a)b\otimes
a^{\prime}b^{\prime}+a\mathrm{d}b\otimes
a^{\prime}b^{\prime}+ab\otimes(\mathrm{d}^{\prime}a^{\prime})b^{\prime}+ab\otimes
a^{\prime}\mathrm{d}^{\prime}b^{\prime}$ $\displaystyle=$
$\displaystyle(\mathrm{d}a\otimes
a^{\prime}+a\otimes\mathrm{d}^{\prime}a^{\prime})\cdot(b\otimes
b^{\prime})+(a\otimes a^{\prime})\cdot(\mathrm{d}b\otimes
b^{\prime}+b\otimes\mathrm{d}^{\prime}b^{\prime})$ $\displaystyle=$
$\displaystyle\mathrm{d}_{A\otimes A^{\prime}}(a\otimes
a^{\prime})\cdot(b\otimes b^{\prime})+(a\otimes
a^{\prime})\cdot\mathrm{d}_{A\otimes A^{\prime}}(b\otimes b^{\prime})$

for all $a,b\in A$ and $a^{\prime},b^{\prime}\in A^{\prime}$. Furthermore,
$\Gamma_{A\otimes A^{\prime}}$ is generated by $A\otimes A^{\prime}$ and
$\mathrm{d}_{A\otimes A^{\prime}}$, since

$\displaystyle(A\otimes A^{\prime})\cdot\mathrm{d}_{A\otimes
A^{\prime}}(A\otimes A^{\prime})=$ $\displaystyle(A\otimes
A^{\prime})\cdot(\mathrm{d}A\otimes A^{\prime}\oplus
A\otimes\mathrm{d}^{\prime}A^{\prime})$ $\displaystyle=$ $\displaystyle
A\mathrm{d}A\otimes A^{\prime}\oplus A\otimes
A^{\prime}\mathrm{d}^{\prime}A^{\prime}$ $\displaystyle=$
$\displaystyle\Gamma\otimes A^{\prime}\oplus A\otimes\Gamma^{\prime}$
$\displaystyle=$ $\displaystyle\Gamma_{A\otimes A^{\prime}}$

as an equation of sets. Finally, associativity follows, since

$\displaystyle\Gamma_{(A\otimes A^{\prime})\otimes A^{\prime\prime}}=$
$\displaystyle\Gamma_{A\otimes A^{\prime}}\otimes
A^{\prime\prime}\oplus(A\otimes A^{\prime})\otimes\Gamma^{\prime\prime}$
$\displaystyle=$ $\displaystyle(\Gamma\otimes A^{\prime}\oplus
A\otimes\Gamma^{\prime})\otimes A^{\prime\prime}\oplus(A\otimes
A^{\prime})\otimes\Gamma^{\prime\prime}$ $\displaystyle=$
$\displaystyle\Gamma\otimes(A^{\prime}\otimes A^{\prime\prime})\oplus
A\otimes(\Gamma^{\prime}\otimes A^{\prime\prime}\oplus
A^{\prime}\otimes\Gamma^{\prime\prime})$ $\displaystyle=$
$\displaystyle\Gamma\otimes(A^{\prime}\otimes A^{\prime\prime})\oplus
A\otimes\Gamma_{A^{\prime}\otimes A^{\prime\prime}}$ $\displaystyle=$
$\displaystyle\Gamma_{A\otimes(A^{\prime}\otimes A^{\prime\prime})}$

as an equation of sets. ∎

######  Remark 3.3.

In other words, Proposition 3.2 proves that FODC form a monoidal category. The
monoidal product is

$(\Gamma,\mathrm{d},A)\otimes(\Gamma^{\prime},\mathrm{d}^{\prime},A^{\prime}):=(\Gamma_{A\otimes
A^{\prime}},\mathrm{d}_{A\otimes A^{\prime}},A\otimes A^{\prime})$ (29)

and the monoidal unit is the trivial calculus $(\Gamma=\\{0\\},\mathrm{d}=0)$
on the algebra $A=\Bbbk$.

Given a FODC on $A$ we can induce a FODC on any subalgebra and quotient
algebra of $A$.

######  Proposition 3.4.

Let $(\Gamma,\mathrm{d})$ be a FODC on $A$.

  1. i.)

An injective algebra map $\iota\colon A^{\prime}\hookrightarrow A$ induces a
FODC $(\Gamma_{\iota},\mathrm{d}_{\iota})$ on $A^{\prime}$, where

$\Gamma_{\iota}:=\iota(A^{\prime})\mathrm{d}\iota(A^{\prime})\subseteq\Gamma$
(30)

and $\mathrm{d}_{\iota}\colon A^{\prime}\ni
a^{\prime}\mapsto\mathrm{d}\iota(a^{\prime})\in\Gamma_{\iota}$.

  2. ii.)

A surjective algebra map $\pi\colon A\rightarrow A^{\prime}$ induces a FODC
$(\Gamma_{\pi},\mathrm{d}_{\pi})$ on $A^{\prime}$, where

$\Gamma_{\pi}:=\Gamma/\Gamma_{I}$ (31)

is the $A$-bimodule quotient111 Equivalence classes in $A^{\prime}=A/I$ are
denoted by $[a]:=a+I$ for a representative $a\in A$, while equivalence classes
in $\Gamma_{\pi}$ are denoted by $[a\mathrm{d}b]:=a\mathrm{d}b+\Gamma_{I}$ for
all $a,b\in A$. with $\Gamma_{I}:=I\mathrm{d}A+A\mathrm{d}I$, where
$I:=\ker\pi\subseteq A$ and $\mathrm{d}_{\pi}\colon
A^{\prime}\ni\pi(a)\mapsto[\mathrm{d}a]\in\Gamma_{\pi}$.

  3. iii.)

If $\iota\colon A^{\prime}\rightarrow A$ and $\pi\colon A\rightarrow
A^{\prime}$ are injective and surjective algebra maps, respectively, such that
$\iota$ is a section of $\pi$, i.e. $\pi\circ\iota=\mathrm{id}_{A^{\prime}}$,
then $(\Gamma_{\iota},\mathrm{d}_{\iota})$ is equivalent to
$(\Gamma_{\pi},\mathrm{d}_{\pi})$. The isomorphism $(\phi,\Phi)$ is given by
$\phi=\mathrm{id}_{A^{\prime}}$ and

$\Phi\colon\Gamma_{\iota}\ni\omega\mapsto[\omega]\in\Gamma_{\pi}.$ (32)

We call $(\Gamma_{\iota},\mathrm{d}_{\iota})$ the pullback calculus, while we
call $(\Gamma_{\pi},\mathrm{d}_{\pi})$ the quotient calculus.

###### Proof.

  1. i.)

The $\Bbbk$-submodule (30) is structured as an $A^{\prime}$-bimodule via

$a^{\prime}\cdot\omega\cdot
b^{\prime}:=\iota(a^{\prime})\omega\iota(b^{\prime})$ (33)

for all $a^{\prime},b^{\prime}\in A^{\prime}$ and
$\omega\in\Gamma_{\iota}\subseteq\Gamma$. To see this, we first verify that
the operations (33) close in $\Gamma_{\iota}$. By definition, any
$\omega\in\Gamma_{\iota}$ is a finite sum $\omega=\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})$, for $a^{\prime i},b^{\prime}_{i}\in
A^{\prime}$. Then, for all $a^{\prime},b^{\prime}\in A^{\prime}$,

$\displaystyle a^{\prime}\cdot\omega\cdot b^{\prime}=$
$\displaystyle\iota(a^{\prime})\iota(a^{\prime
i})(\mathrm{d}\iota(b^{\prime}_{i}))\iota(b^{\prime})$ $\displaystyle=$
$\displaystyle\iota(a^{\prime}a^{\prime
i})(\mathrm{d}(\iota(b^{\prime}_{i})\iota(b^{\prime}))-\iota(b^{\prime}_{i})\mathrm{d}\iota(b^{\prime}))$
$\displaystyle=$ $\displaystyle\iota(a^{\prime}a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i}b^{\prime})-\iota(a^{\prime}a^{\prime
i}b^{\prime}_{i})\mathrm{d}\iota(b^{\prime})\in\Gamma_{\iota},$

where we used that $\iota$ is an algebra map. The operations (31) are
$A^{\prime}$-coactions because $\iota$ is an algebra map and they are clearly
commuting. Now $\mathrm{d}_{\iota}$ is a well-defined $\Bbbk$-linear map,
satisfying

$\displaystyle\mathrm{d}_{\iota}(a^{\prime}b^{\prime})=\mathrm{d}\iota(a^{\prime}b^{\prime})=\mathrm{d}(\iota(a^{\prime})\iota(b^{\prime}))=(\mathrm{d}\iota(a^{\prime}))\iota(b^{\prime})+\iota(a^{\prime})\mathrm{d}\iota(b^{\prime})=(\mathrm{d}_{\iota}a^{\prime})\cdot
b^{\prime}+a^{\prime}\cdot\mathrm{d}_{\iota}b^{\prime}$

for all $a^{\prime},b^{\prime}\in A^{\prime}$. We already noticed that any
$\omega\in\Gamma_{\iota}$ is of the form $\omega=\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})$ for some $a^{\prime i},b^{\prime}_{i}\in
A^{\prime}$. Then $a^{\prime
i}\cdot\mathrm{d}_{\iota}b^{\prime}_{i}=\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})=\omega$, proving that
$(\Gamma_{\iota},\mathrm{d}_{\iota})$ is a FODC on $A^{\prime}$.

  2. ii.)

First note that $I=\ker\pi\subseteq A$ is an algebra ideal. Then, by the
Leibniz rule of $\mathrm{d}$, it immediately follows that
$\Gamma_{I}=I\mathrm{d}A+A\mathrm{d}I\subseteq\Gamma$ is an $A$-subbimodule.
Hence, (31) is an $A$-bimodule. Since $I\cdot\Gamma,\Gamma\cdot
I\subseteq\Gamma_{I}$ the $A$-bimodule actions on $\Gamma_{\pi}$ descend to an
$A^{\prime}=A/I$-bimodule action

$\pi(a)\cdot[\omega]\cdot\pi(b):=[a\omega b]$ (34)

for all $a,b\in A$ and $\omega\in\Gamma$. Then, $\mathrm{d}_{\pi}\colon
A^{\prime}\ni\pi(a)\mapsto[\mathrm{d}a]\in\Gamma_{\pi}$ is $\Bbbk$-linear and
satisfies the Leibniz rule

$\displaystyle\mathrm{d}_{\pi}(\pi(a)\pi(b))=$
$\displaystyle\mathrm{d}_{\pi}(\pi(ab))=[\mathrm{d}(ab)]=[(\mathrm{d}a)b+a\mathrm{d}b]=[(\mathrm{d}a)b]+[a\mathrm{d}b]$
$\displaystyle=$
$\displaystyle[\mathrm{d}a]\cdot\pi(b)+\pi(a)\cdot[\mathrm{d}b]=(\mathrm{d}_{\pi}\pi(a))\cdot\pi(b)+\pi(a)\cdot\mathrm{d}_{\pi}(\pi(b))$

for all $a,b\in A$. Finally, $A^{\prime}$ and $\mathrm{d}_{\pi}$ generate
$\Gamma_{\pi}$, since for all $\omega\in\Gamma$ there are $a^{i},b_{i}\in A$
such that $\omega=a^{i}\mathrm{d}b_{i}$ and thus
$[\omega]=[a^{i}\mathrm{d}b_{i}]=\pi(a^{i})\cdot\mathrm{d}_{\pi}(\pi(b_{i}))$.
We conclude that $(\Gamma_{\pi},\mathrm{d}_{\pi})$ is a FODC on $A^{\prime}$.

  3. iii.)

According to i.) and ii.) $(\Gamma_{\iota},\mathrm{d}_{\iota})$ and
$(\Gamma_{\pi},\mathrm{d}_{\pi})$ are FODC on $A^{\prime}$. The $\Bbbk$-linear
map (32) explicitly reads

$\displaystyle\Gamma_{\iota}\ni\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})\mapsto[\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})]=\pi(\iota(a^{\prime
i}))\cdot\mathrm{d}_{\pi}(\pi(\iota(b^{\prime}_{i})))=a^{\prime
i}\cdot\mathrm{d}_{\pi}(b^{\prime}_{i})\in\Gamma_{\pi},$

where we used $\pi\circ\iota=\mathrm{id}_{A^{\prime}}$ and that any
$\omega\in\Gamma_{\iota}$ is a finite sum $\omega=\iota(a^{\prime
i})\mathrm{d}\iota(b^{\prime}_{i})\in\Gamma_{\iota}\subseteq\Gamma$ for
$a^{\prime i},b^{\prime}_{i}\in A^{\prime}$. The inverse of (32) is then
clearly given by $a^{\prime
i}\cdot\mathrm{d}_{\pi}(b^{\prime}_{i})\in\Gamma_{\pi}\mapsto\iota(a^{\prime
i})\cdot\mathrm{d}\iota(b^{\prime}_{i})\in\Gamma_{\iota}$. In particular

$\Phi(\mathrm{d}_{\iota}(a^{\prime}))=\Phi(\mathrm{d}\iota(a^{\prime}))=[\mathrm{d}\iota(a^{\prime})]=\mathrm{d}_{\pi}(\pi(\iota(a^{\prime})))=\mathrm{d}_{\pi}(a^{\prime})$

for all $a^{\prime}\in A^{\prime}$. Further observe that
$\Phi(a^{\prime}\cdot\omega\cdot b^{\prime})=[a^{\prime}\cdot\omega\cdot
b^{\prime}]=\pi(a^{\prime})\cdot[\omega]\cdot\pi(b^{\prime})=\pi(a^{\prime})\cdot\Phi(\omega)\cdot\pi(b^{\prime})$
for all $a^{\prime},b^{\prime}\in A^{\prime}$ and $\omega\in\Gamma_{\iota}$ by
definition of the $A^{\prime}$-bimodule actions.

This concludes the proof of the proposition. ∎

######  Remark 3.5.

In Proposition 3.4 i.) the injectivity assumption is not needed in order to
obtain a FODC. The reason is that any algebra map $\iota\colon
A^{\prime}\rightarrow A$ induces an injective algebra map
$\iota^{\prime}\colon A^{\prime}/\ker\iota\rightarrow A$. By Proposition 3.4
i.) this induces a FODC
$(\Gamma_{\iota^{\prime}},\mathrm{d}_{\iota^{\prime}})$ on
$A^{\prime}/\ker\iota$. This calculus can be understood as a FODC on
$A^{\prime}$ because $\ker\iota$ acts trivially on $\Gamma_{\iota^{\prime}}$
and $\mathrm{d}_{\iota^{\prime}}$ vanishes on $\ker\iota$.

###  3.2 Covariant calculi on comodule algebras

In this section we discuss the theory for covariant calculi on comodule
algebras, generalizing the perspectives of [29, 55], see also [47]. Let $H$ be
a $\Bbbk$-Hopf algebra.

######  Definition 3.6.

  1. i.)

A FODC $(\Gamma,\mathrm{d})$ on a right $H$-comodule algebra $(A,\delta_{R})$
is said to be right $H$-covariant if

$a\mathrm{d}a^{\prime}\mapsto a_{0}\mathrm{d}a^{\prime}_{0}\otimes
a_{1}a^{\prime}_{1},$ (35)

for all $a,a^{\prime}\in A$, extends to a well-defined $\Bbbk$-linear map
$\Delta_{R}\colon\Gamma\rightarrow\Gamma\otimes H$.

  2. ii.)

A FODC $(\Gamma,\mathrm{d})$ on a left $H$-comodule algebra $(A,\delta_{L})$
is said to be left $H$-covariant if

$a\mathrm{d}a^{\prime}\mapsto a_{-1}a^{\prime}_{-1}\otimes
a_{0}\mathrm{d}a^{\prime}_{0},$ (36)

for all $a,a^{\prime}\in A$, extends to a well-defined $\Bbbk$-linear map
$\Delta_{L}\colon\Gamma\rightarrow H\otimes\Gamma$.

  3. iii.)

A FODC $(\Gamma,\mathrm{d})$ on an $H$-bicomodule algebra
$(A,\delta_{R},\delta_{L})$ is said to be $H$-bicovariant if it is right and
left $H$-covariant.

######  Lemma 3.7.

  1. i.)

Let $(\Gamma,\mathrm{d})$ be a right $H$-covariant FODC on $(A,\delta_{R})$.
Then (35) defines a right $H$-coaction $\Delta_{R}$ on $\Gamma$ such that

$\Delta_{R}(a\omega
a^{\prime})=\delta_{R}(a)\Delta_{R}(\omega)\delta_{R}(a^{\prime})$ (37)

holds for all $a,a^{\prime}\in A$ and $\omega\in\Gamma$, i.e. $\Gamma$ is a
right $H$-covariant $A$-bimodule.

  2. ii.)

If $(A,\delta_{L})$ is a left $H$-comodule algebra and $(\Gamma,\mathrm{d})$ a
left $H$-covariant FODC on $A$, (36) is a left $H$-coaction on $\Gamma$ such
that

$\Delta_{L}(a\omega
a^{\prime})=\delta_{L}(a)\Delta_{L}(\omega)\delta_{L}(a^{\prime})$ (38)

for all $a,a^{\prime}\in A$ and $\omega\in\Gamma$. This means that $\Gamma$ is
a left $H$-covariant $A$-bimodule.

  3. iii.)

In the case of $(A,\delta_{R},\delta_{L})$ being an $H$-bicomodule algebra and
$(\Gamma,\mathrm{d})$ an $H$-bicovariant FODC on $A$ the two $H$-coactions
$\Delta_{R}$ and $\Delta_{L}$ commute and (37), (38) hold. In other words,
$\Gamma$ is an $H$-bicovariant $A$-bimodule.

###### Proof.

Assume that (35) is a well-defined $\Bbbk$-linear map and let $a,a^{\prime}\in
A$ and $\omega\in\Gamma$ be arbitrary. There are $f^{i},g^{i}\in A$ such that
$\omega=\sum_{i}f^{i}\mathrm{d}g^{i}$. Then

$\displaystyle\delta_{R}(a)\Delta_{R}(\omega)\delta_{R}(a^{\prime})=$
$\displaystyle
a_{0}\sum_{i}f^{i}_{0}\mathrm{d}(g^{i}_{0})a^{\prime}_{0}\otimes
a_{1}f^{i}_{1}g^{i}_{1}a^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\sum_{i}a_{0}f^{i}_{0}(\mathrm{d}(g^{i}_{0}a^{\prime}_{0})-g^{i}_{0}\mathrm{d}a^{\prime}_{0})\otimes
a_{1}f^{i}_{1}g^{i}_{1}a^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\sum_{i}(af^{i})_{0}\mathrm{d}((g^{i}a^{\prime})_{0})\otimes(af^{i})_{1}(g^{i}a^{\prime})_{1}-\sum_{i}(af^{i}g^{i})_{0}\mathrm{d}a^{\prime}_{0}\otimes(af^{i}g^{i})_{1}a^{\prime}_{1}$
$\displaystyle=$
$\displaystyle\sum_{i}\Delta_{R}(af^{i}\mathrm{d}(g^{i}a^{\prime}))-\sum_{i}\Delta_{R}(af^{i}g^{i}\mathrm{d}a^{\prime})$
$\displaystyle=$
$\displaystyle\Delta_{R}\bigg{(}\sum_{i}af^{i}(\mathrm{d}(g^{i}a^{\prime})-g^{i}\mathrm{d}a^{\prime})\bigg{)}$
$\displaystyle=$
$\displaystyle\Delta_{R}\bigg{(}\sum_{i}af^{i}\mathrm{d}(g^{i})a^{\prime}\bigg{)}$
$\displaystyle=$ $\displaystyle\Delta_{R}(a\omega a^{\prime}),$

where we used the Leibniz, the definition (35) of $\Delta_{R}$ and that $A$ is
a right $H$-comodule algebra. It is left to prove that $\Delta_{R}$ is a right
$H$-coaction on $\Gamma$, which is the case, since

$\displaystyle(\Delta_{R}\otimes\mathrm{id}_{H})(\Delta_{R}(a\mathrm{d}a^{\prime}))=$
$\displaystyle\Delta_{R}(a_{0}\mathrm{d}a^{\prime}_{0})\otimes
a_{1}a^{\prime}_{1}$ $\displaystyle=$
$\displaystyle(a_{0})_{0}\mathrm{d}((a^{\prime}_{0})_{0})\otimes(a_{0})_{1}(a^{\prime}_{0})_{1}\otimes
a_{1}a^{\prime}_{1}$ $\displaystyle=$ $\displaystyle
a_{0}\mathrm{d}a^{\prime}_{0}\otimes(a_{1})_{1}(a^{\prime}_{1})_{1}\otimes(a_{1})_{2}(a^{\prime}_{1})_{2}$
$\displaystyle=$ $\displaystyle
a_{0}\mathrm{d}a^{\prime}_{0}\otimes(a_{1}a^{\prime}_{1})_{1}\otimes(a_{1}a^{\prime}_{1})_{2}$
$\displaystyle=$
$\displaystyle(\mathrm{id}_{\Gamma}\otimes\Delta)(\Delta_{R}(a\mathrm{d}a^{\prime}))$

for all $a,a^{\prime}\in A$, where we used that $\delta_{R}$ is a right
$H$-coaction and that the coproduct $\Delta$ is an algebra homomorphism. The
left covariant case follows in complete analogy. Assume now that $A$ is an
$H$-bicomodule algebra and $(\Gamma,\mathrm{d})$ $H$-bicovariant. It remains
to prove that the $H$-coactions $\Delta_{R}$ and $\Delta_{L}$ commute. For all
$a,a^{\prime}\in A$

$\displaystyle(\Delta_{L}\otimes\mathrm{id}_{H})(\Delta_{R}(a\mathrm{d}a^{\prime}))=$
$\displaystyle\Delta_{L}(a_{0}\mathrm{d}a^{\prime}_{0})\otimes
a_{1}a^{\prime}_{1}$ $\displaystyle=$
$\displaystyle(a_{0})_{-1}(a^{\prime}_{0})_{-1}\otimes(a_{0})_{0}\mathrm{d}((a^{\prime}_{0})_{0})\otimes
a_{1}a^{\prime}_{1}$ $\displaystyle=$ $\displaystyle
a_{-1}a^{\prime}_{-1}\otimes(a_{0})_{0}\mathrm{d}((a^{\prime}_{0})_{0})\otimes(a_{0})_{1}(a^{\prime}_{0})_{1}$
$\displaystyle=$ $\displaystyle
a_{-1}a^{\prime}_{-1}\otimes\Delta_{R}(a_{0}\mathrm{d}(a^{\prime}_{0}))$
$\displaystyle=$
$\displaystyle(\mathrm{id}_{H}\otimes\Delta_{R})(\Delta_{L}(a\mathrm{d}a^{\prime}))$

follows since $\delta_{R}$ and $\delta_{L}$ are commuting $H$-coactions on
$A$. ∎

The right $H$-coaction $\Delta_{R}$ in (3.6) is defined from the right
$H$-coaction $\delta_{R}$ on $A$ in such a way that the differential
$\mathrm{d}$ is right $H$-colinear. Note that there is a weaker notion of
right $H$-covariant FODC on $A$ if we replace (3.6) by the request that there
is a right $H$-coaction $\Delta_{R}\colon\Gamma\rightarrow\Gamma\otimes H$ on
$\Gamma$ such that (37) holds. We are mainly interested in (3.6), because
$\Gamma$ is given by the data of $A$ and $\mathrm{d}$, so it is natural to ask
the coaction on $\Gamma$ to arise from the coaction on $A$, as well.

In the light of Lemma 3.7 we define a morphism of right/left/bi $H$-covariant
FODC $(\Gamma,\mathrm{d})$ and $(\Gamma^{\prime},\mathrm{d}^{\prime})$ on
right/left/bi $H$-comodule algebras $A,A^{\prime}$ to be a morphism
$(\phi,\Phi)$ between FODC such that $\phi$ and $\Phi$ are right/left/bi
$H$-colinear maps. The natural question arises if Proposition 3.4 generalizes
to the covariant setting.

######  Proposition 3.8.

Let $(\Gamma,\mathrm{d})$ be a right $H$-covariant FODC on a right
$H$-comodule algebra $A$.

  1. i.)

If $\iota\colon B\hookrightarrow A$ is an injective right $H$-comodule algebra
homomorphism the pullback calculus $(\Gamma_{\iota},\mathrm{d}_{\iota})$ is a
right $H$-covariant FODC on $B$.

  2. ii.)

If $\pi\colon A\rightarrow A^{\prime}$ is a surjective induces right
$H$-comodule algebra homomorphism the quotient calculus
$(\Gamma_{\pi},\mathrm{d}_{\pi})$ is a right $H$-covariant FODC on
$A^{\prime}$.

  3. iii.)

If $\iota\colon A^{\prime}\rightarrow A$ and $\pi\colon A\rightarrow
A^{\prime}$ are injective and surjective right $H$-comodule algebra
homomorphisms, respectively, such that $\iota$ is a section of $\pi$, then
$(\Gamma_{\iota},\mathrm{d}_{\iota})$ is equivalent to
$(\Gamma_{\pi},\mathrm{d}_{\pi})$ as a right $H$-covariant FODC.

Analogous statements hold for left $H$-covariant and $H$-bicovariant FODC on
left $H$-comodule algebras and $H$-bicomodule algebras, respectively.

###### Proof.

From Proposition 3.4 it follows that $(\Gamma_{\iota},\mathrm{d}_{\iota})$ and
$(\Gamma_{\pi},\mathrm{d}_{\pi})$ are FODC on $B$ and $A^{\prime}$,
respectively. We observe that for all $f^{i},g^{i}\in A$

$\iota(f^{i})_{0}\mathrm{d}(\iota(g^{i})_{0})\otimes\iota(f^{i})_{1}\iota(g^{i})_{1}=\iota(f^{i}_{0})\mathrm{d}(\iota(g^{i}_{0}))\otimes
f^{i}_{1}g^{i}_{1}\in\Gamma_{\iota}\otimes H$

by the right $H$-colinearity of $\iota$. Thus, the right $H$-coaction
$\Delta_{R}\colon\Gamma\rightarrow\Gamma\otimes H$ restricts to a
$\Bbbk$-linear map
$\Delta_{R}|_{\Gamma_{\iota}}\colon\Gamma_{\iota}\rightarrow\Gamma_{\iota}\otimes
H$ and consequently $(\Gamma_{\iota},\mathrm{d}_{\iota})$ is a right
$H$-covariant FODC. For $(\Gamma_{\pi},\mathrm{d}_{\pi})$ we first note that
$I=\ker\pi\subseteq A$ satisfies $\delta_{R}(I)\subseteq I\otimes H$ since
$\pi$ is right $H$-colinear. Then, by the right $H$-covariance of
$(\Gamma,\mathrm{d})$ $\Delta_{R}(\Gamma_{I})\subseteq\Gamma_{I}$ and
$\Delta_{R}\colon\Gamma\rightarrow\Gamma\otimes H$ descends to a well-defined
right $H$-coaction on $\Gamma_{\pi}$. It reads

$\Delta_{R}([a\mathrm{d}a^{\prime}])=[a_{0}\mathrm{d}a^{\prime}_{0}]\otimes
a_{1}a^{\prime}_{1}=\pi(a_{0})\cdot\mathrm{d}_{\pi}\pi(a^{\prime}_{0})\otimes
a_{1}a^{\prime}_{1}$

for all $a,a^{\prime}\in A$ and thus $(\Gamma_{\pi},\mathrm{d}_{\pi})$ is
right $H$-covariant. For the third statement note that (32) is clearly right
$H$-colinear. ∎

###  3.3 Bicovariant calculi

In his seminal work [55], Woronowicz introduced a FODC on Hopf algebras with
the additional property that the coproduct induces coactions on the module of
differential forms. This is a special case of the covariant calculi on
comodule algebras which we discussed in the previous section. Since such
bicovariant calculi appear in our examples we direct some attention to them in
this section.

######  Definition 3.9.

A FODC $(\Gamma,\mathrm{d})$ on a Hopf algebra $H$ is said to be bicovariant
if it is $H$-bicovariant where we endow $H$ with the right and left
$H$-coaction given by the coproduct.

Explicitly, a bicovariant FODC $(\Gamma,\mathrm{d})$ on $H$ is a FODC on $H$
such that

$\Delta_{R}\colon\Gamma\ni\sum\nolimits_{i}a^{i}\mathrm{d}b^{i}\mapsto\sum\nolimits_{i}a^{i}_{1}\mathrm{d}b^{i}_{1}\otimes
a^{i}_{2}b^{i}_{2}\in\Gamma\otimes H$ (39)

and

$\Delta_{L}\colon\Gamma\ni\sum\nolimits_{i}a^{i}\mathrm{d}b^{i}\mapsto\sum\nolimits_{i}a^{i}_{1}b^{i}_{1}\otimes
a^{i}_{2}\mathrm{d}b^{i}_{2}\in H\otimes\Gamma$ (40)

are well-defined maps. According to Lemma 3.7 $\Delta_{R}$ and $\Delta_{L}$
are commuting right and left $H$-coactions on $\Gamma$, respectively, i.e.
$(\Gamma,\Delta_{R},\Delta_{L})$ is an $H$-bicovariant $H$-bimodule. Note that
by definition, the differential is $H$-colinear with respect to the coactions,
i.e. $\Delta_{R}(\mathrm{d}a)=\mathrm{d}a_{1}\otimes a_{2}$ and
$\Delta_{L}(\mathrm{d}a)=a_{1}\otimes\mathrm{d}a_{2}$ for all $a\in H$.

Consider the following construction of bicovariant FODC on $H$: let
$\mathcal{R}\subseteq\ker\epsilon\subseteq H$ be a right ideal and define
$\mathcal{N}:=\chi^{-1}(H\otimes\mathcal{R})\subseteq H\otimes H$, where

$\chi\colon H\otimes H\ni a\otimes a^{\prime}\mapsto aa^{\prime}_{1}\otimes
a^{\prime}_{2}\in H\otimes H$ (41)

is the canonical map with inverse $\chi^{-1}(a\otimes
a^{\prime})=aS(a^{\prime}_{1})\otimes a^{\prime}_{2}$. It is easy to verify
that $\mathcal{N}\subseteq\ker\mu_{H}$ is a $H$-subbimodule and a left
$H$-comodule. Consequently, the universal calculus
$(\ker\mu_{H},\mathrm{d}_{u})$ descends to a left $H$-covariant FODC
$(\ker\mu_{H}/\mathcal{N},\mathrm{d})$ on $H$, where
$\mathrm{d}:=\pi\circ\mathrm{d}_{u}$ and
$\pi\colon\ker\mu_{H}\rightarrow\ker\mu_{H}/\mathcal{N}$ is the quotient map.
If $\mathcal{R}$ is $\mathrm{ad}$-invariant, i.e.
$\mathrm{ad}(\mathcal{R})\subseteq\mathcal{R}\otimes H$ for $\mathrm{ad}\colon
H\ni a\mapsto a_{2}\otimes S(a_{1})a_{3}\in H\otimes H$, it follows that
$(\ker\mu_{H}/\mathcal{N},\mathrm{d})$ is a bicovariant FODC on $H$. It turn
out that this construction classifies bicovariant FODC on $H$.

######  Theorem 3.10 (Woronowicz).

For any $\mathrm{ad}$-invariant right ideal
$\mathcal{R}\subseteq\ker\epsilon\subseteq H$ there is a bicovariant FODC
$(\ker\mu_{H}/\mathcal{N},\mathrm{d})$ on $H$, where
$\mathcal{N}:=\kappa^{-1}(H\otimes\mathcal{R})$ and
$\mathrm{d}:=\pi\circ\mathrm{d}_{u}$. Every bicovariant FODC on $H$ is of this
form.

In particular, for every Hopf algebra $H$ we can consider the bicovariant FODC
constructed from the $\mathrm{ad}$-invariant right ideals $\\{0\\}$ and
$\ker\epsilon$. The first one corresponds to the universal calculus, a kind of
maximal choice, while the second one is the smallest quotient, i.e. the other
extreme. When we refer to a bicovariant FODC on $H$ we always assume
implicitly the choice of an $\mathrm{ad}$-invariant right ideal
$\mathcal{R}\subseteq\ker\epsilon$.

Let $(\Gamma,\mathrm{d})$ be a bicovariant FODC on $H$. By Theorem 2.4 we can
choose a basis $\\{\omega^{i}\\}_{i\in I}$ of ${}^{\mathrm{co}H}\Gamma$ and
for every $\omega\in\Gamma$ there are unique $a_{i}\in H$ such that
$\omega=a_{i}\omega^{i}$. As a further consequence [55], there are functionals
$f^{i}{}_{j}\colon H\rightarrow\Bbbk$ such that

$\omega^{i}a=(f^{i}{}_{j}*a)\omega^{j}$ (42)

for all $a\in H$, where $f^{i}{}_{j}*a:=a_{1}(f^{i}{}_{j}(a_{2}))$ is the
convolution action. Those functionals tell us how to commute elements of the
Hopf algebra with elements of ${}^{\mathrm{co}H}\Gamma$.

There is a canonical way to construct a right covariant differential calculus
on quantum groups viewed as comodule with respect to one of its quantum
subgroups.

######  Example 3.11.

Let $A$ be a Hopf algebra, $I\subseteq A$ a Hopf ideal and $\pi\colon
A\rightarrow H:=A/I$ the quotient Hopf algebra projection. Choose a
bicovariant FODC $(\Gamma,\mathrm{d})$ on $A$. We define the right
$H$-coactions

$\delta_{R}:=(\mathrm{id}_{A}\otimes\pi)\circ\Delta\colon A\ni f\mapsto
f_{0}\otimes f_{1}\in A\otimes H$ (43)

and

$\Delta_{R}\colon\Gamma\ni f\mathrm{d}g\mapsto f_{0}\mathrm{d}g_{0}\otimes
f_{1}g_{1}\in\Gamma\otimes H.$ (44)

In fact (44), is well-defined: let $f^{i},g^{i}\in A$ such that
$\sum_{i}f^{i}\mathrm{d}g^{i}=0$. Then
$\sum_{i}f^{i}_{1}\mathrm{d}g^{i}_{1}\otimes f^{i}_{2}g^{i}_{2}=0$, since
$(\Gamma,\mathrm{d})$ is a bicovariant FODC on $A$. It follows that

$0=\sum_{i}f^{i}_{1}\mathrm{d}g^{i}_{1}\otimes\pi(f^{i}_{2}g^{i}_{2})=\sum_{i}f^{i}_{1}\mathrm{d}g^{i}_{1}\otimes\pi(f^{i}_{2})\pi(g^{i}_{2})=\sum_{i}f^{i}_{0}\mathrm{d}g^{i}_{0}\otimes
f^{i}_{1}g^{i}_{1}$

and consequently (44) is well-defined. By Lemma 3.7, $\Delta_{R}$ is a right
$H$-coaction on $\Gamma$ such that (37) holds. We further induce a right
$H$-covariant FODC $(\Gamma_{B},\mathrm{d}_{B})$ on the subalgebra
$B:=A^{\mathrm{co}H}\subseteq A$ by Proposition 3.4. Note that $B$ might be
non-trivial, while $A^{\mathrm{co}A}=\Bbbk\cdot 1_{A}$ only admits the trivial
calculus on it.

####  3.3.1 The Example of $\mathrm{SL}_{q}(2)$ and its quotient
$\mathcal{O}_{q}(P)$

We now want to elucidate the results of Proposition 3.8 by taking as $A$ the
Manin deformation (see [38]) of the ring of algebraic functions on the complex
special linear group $\mathrm{SL}_{2}(\mathbb{C})$ and as the FODC on $A$ the
$4D_{+}$ bicovariant calculus studied by Woronowicz [57].

Consider $\mathcal{O}_{q}(\mathrm{SL}_{2})$, the deformed algebra of regular
functions on the complex special linear group $\mathrm{SL}_{2}(\mathbb{C})$;
it is freely generated over $\mathbb{C}_{q}:=\mathbb{C}[q,q^{-1}]$ by the
symbols $a_{\mu\nu}=\left({}_{\gamma}^{\alpha}~{}{}_{\delta}^{\beta}\right)$,
modulo the ideal $I_{M}$

$\begin{array}[]{c}\alpha\beta=q^{-1}\beta\alpha,\quad\alpha\gamma=q^{-1}\gamma\alpha,\quad\beta\delta=q^{-1}\delta\beta,\quad\gamma\delta=q^{-1}\delta\gamma,\\\
\\\
\beta\gamma=\gamma\beta\qquad\alpha\delta-\delta\alpha=(q^{-1}-q)\beta\gamma\end{array}$

and modulo the ideal $(\alpha\delta-q^{-1}\beta\gamma-1)$ generated by the
determinant relation. In short:

$A:=\mathcal{O}_{q}(\mathrm{SL}_{2})\,=\,\mathbb{C}_{q}\langle\alpha,\beta,\gamma,\delta\rangle/I_{M}+(\alpha\delta-q^{-1}\beta\gamma-1)~{}.$

It is well know that $A$ is an Hopf algebra where the comultiplication is
given by $\Delta a_{\mu\nu}=a_{\mu\rho}\otimes a_{\rho\nu}$. Let
$H:=\mathcal{O}_{q}(P)$ be the deformed algebra of functions on the Borel
subgroup $P\subset\mathrm{SL}_{2}(\mathbb{C})$. We identify it with the
quotient $\mathcal{O}_{q}(\mathrm{SL}_{2})/I_{P}$ with $I_{P}$ being the Hopf
ideal generated by the element $\gamma$. Denoting by $t$ and $p$ the
equivalence classes $[\alpha]$ and $[\beta]$ defined by the equivalence
relation given by $\sigma\sim\sigma+\iota$ with $\sigma\in A$ and $\iota\in
I_{P}$ and setting $[\gamma]=0$, we get

$\mathcal{O}_{q}(P)=\mathbb{C}_{q}\langle
t,t^{-1},p\rangle/(tp-q^{-1}pt)=\mathbb{C}_{q}\langle
t,s,p\rangle/(ts-1,st-1,tp-q^{-1}pt)~{}.$

Consider then the projection

$\pi:\mathcal{O}_{q}(\mathrm{SL}_{2})\longrightarrow\mathcal{O}_{q}(\mathrm{SL}_{2})/(I_{P})~{}~{}$

that on the generators reads
$\left({}_{\gamma}^{\alpha}~{}{}_{\delta}^{\beta}\right)\mapsto\left({}_{0}^{t}~{}{}_{t^{-1}}^{p}\right)$
and is extended as an algebra map. We can then define the right $H$ coaction
on $A$ by

$\delta_{R}=(id\otimes\pi)\Delta:\mathcal{O}_{q}(\mathrm{SL}_{2})\rightarrow\mathcal{O}_{q}(\mathrm{SL}_{2})\otimes{\mathcal{O}_{q}}(P)~{}.$
(45)

It is well known that (see for example [34] Section 3.1.2) when $q^{4}\neq
0,1$ the deformation of the $\mathfrak{sl}(2)$ envelopping algebra, namely
$\mathcal{U}_{q}\mathfrak{sl}(2)$, is the Hopf algebra freely generated by the
symbols $\\{E,K,K^{-1},F\\}$ with the algebra and coalgebra structure captured
by

$\begin{array}[]{ccc}KE=q^{-1}EK,&KK^{-1}=1=K^{-1}K,&KF=qFK,\\\\[14.22636pt]
&EF-FE=\frac{K^{2}-K^{-2}}{\lambda}&\end{array}$

and

$\begin{array}[]{ccc}\Delta E=E\otimes K+K^{-1}\otimes E\,,&\Delta K^{\pm
1}=K^{\pm 1}\otimes K^{\pm 1}\,,&\Delta F=F\otimes K+K^{-1}\otimes
F,\\\\[14.22636pt] S(E)=-q^{-1}E,&S(K)=K^{-1},&S(F)=-qF,\\\\[14.22636pt]
\epsilon(E)=0,&\epsilon(K)=1,&\epsilon(F)=0.\end{array}$

The duality between $\mathcal{U}_{q}\mathfrak{sl}(2)$ and
$\mathcal{O}_{q}(\mathrm{SL}_{2})$ is explicitly given by

$K(a)=q^{\frac{1}{2}}\,,\,\,K(d)=q^{-\frac{1}{2}}\,,\,E(c)=1=F(b).$

On the Hopf algebra $\mathcal{O}_{q}(\mathrm{SL}_{2})$ one can define two
different bicovariant FODC (see for example [34] Section 14.2.4). We choose
the $4D_{+}$ calculus, defined on generators by the following relations

$\begin{array}[]{rcl}\mathrm{d}\alpha&=&\left(\frac{q-1}{\lambda}\right)\alpha\omega^{1}+\left(\frac{q^{-1}-1}{\lambda}\right)\alpha\omega^{4}-\beta\omega^{2}\\\\[8.53581pt]
\mathrm{d}\gamma&=&\left(\frac{q-1}{\lambda}\right)\gamma\omega^{1}+\left(\frac{q^{-1}-1}{\lambda}\right)\gamma\omega^{4}-\delta\omega^{2}\\\\[8.53581pt]
\mathrm{d}\beta&=&Q\beta\omega^{1}+\left(\frac{q-1}{\lambda}\right)\beta\omega^{4}-\alpha\omega^{3}\\\\[8.53581pt]
\mathrm{d}\delta&=&Q\delta\omega^{1}+\left(\frac{q-1}{\lambda}\right)\delta\omega^{4}-\gamma\omega^{3}\end{array}$

with $\lambda:=q^{-1}-q$ and $Q:=\frac{\lambda^{2}(q^{-1}+1)-1}{\lambda}$ and
where $\\{\omega^{i}\\}_{{}_{i=1,..,4}}$ denote as usual the base of left
invariant $1$-forms. Moreover, we know (see Section 3.3) that the right and
left $A$ module action on $\Gamma_{A}$ are related by the relation
$\omega^{i}\ell=\sum_{{}_{i=1,..,4}}\left(f^{i}_{\,\,j}*\ell\right)\omega^{j}$
where $\ell$ is a general element in $A$; it is convenient to collect the
$f^{i}_{\,\,j}$ functionals in a matrix as follows

$f^{i}_{\,\,j}=\left(\begin{array}[]{cccc}K^{2}&0&0&0\\\ -q^{-3/2}\lambda
KF&\epsilon&0&0\\\ -q^{-3/2}\lambda EK&0&\epsilon&0\\\
q^{-1}\lambda^{2}EF&\lambda\chi_{2}&\lambda\chi_{3}&K^{-2}\end{array}\right)$

using which we get the following useful informations:

$\begin{array}[]{rclcrcl}\omega^{1}\alpha&=&q\alpha\omega^{1}&&\omega^{1}\gamma&=&q\gamma\omega^{1}\\\\[8.53581pt]
\omega^{2}\alpha&=&\alpha\omega^{2}&&\omega^{2}\gamma&=&\gamma\omega^{2}\\\\[8.53581pt]
\omega^{3}\alpha&=&\alpha\omega^{3}-q^{-1}\lambda\beta\omega^{1}&&\omega^{3}\gamma&=&\gamma\omega^{3}-q^{-1}\lambda\delta\omega^{1}\\\\[8.53581pt]
\omega^{4}\alpha&=&q^{-1}\alpha\omega^{4}-\lambda\beta\omega^{2}&&\omega^{4}\gamma&=&q^{-1}\gamma\omega^{4}-\lambda\delta\omega^{2}\end{array}$
(46)

We now restrict our attention to the calculus induced by
$(\Gamma_{A},\mathrm{d})$ on $\mathcal{O}_{q}(P)$. We denote by $\Xi$ a
general element of $\Gamma_{I_{P}}:=I_{P}\mathrm{d}A+A\mathrm{d}I_{P}$ and
$\underline{\rho}=[\rho]$ is the equivalence class defined by the relation
$\rho\sim\rho+\Xi$. Observe then that

$q^{-1}\gamma\mathrm{d}\alpha-\alpha\mathrm{d}\gamma=\omega^{2}\,\,\,\,\,\,\rightarrow\,\,\,\,\,\underline{\omega}^{2}=0.$

Moreover we note that

$[\mathrm{d}(\alpha\delta)]=[\delta(\mathrm{d}\alpha)]+[(\mathrm{d}\delta)\alpha]=\bigg{[}\left(\frac{q-1}{\lambda}+qQ\right)\omega^{1}\bigg{]}$

where we used that $\omega^{1}\alpha=q\alpha\omega^{1}$,
$\omega^{4}\alpha=q^{-1}\alpha\omega^{4}+\Xi$. Using then the determinant
relation $\alpha\delta=1-q^{-1}\beta\gamma$ that implies
$\mathrm{d}(\alpha\delta)\in\Gamma_{I_{P}}$, we obtain
$\underline{\omega}^{1}=0$ Indeed in the end we obtain the following
bicovariant FODC 222Note that a different choice for the FODC on $A$ yields to
a slightly different caclulus $H$ in this case where
$\mathrm{d}t=-\left(\frac{q^{-1}+1}{\lambda}\right)t\underline{\omega}^{4}$
and
$\mathrm{d}p=t\,\underline{\omega}^{3}-\left(\frac{q+1}{\lambda}\right)p\,\underline{\omega}^{4}$.
on $\mathcal{O}_{q}(P)$

$\mathrm{d}t=\left(\frac{q^{-1}-1}{\lambda}\right)t\,\underline{\omega}^{4}\,,\,,\,\,\,\\\
\mathrm{d}p=-t\,\underline{\omega}^{3}+\left(\frac{q-1}{\lambda}\right)p\,\underline{\omega}^{4}$

with

$\underline{\omega}^{3}t=t\underline{\omega}^{3}\,,\,\,\widehat{\omega}^{3}p=p\underline{\omega}^{3}\,,\,\,\underline{\omega}^{4}t=t\underline{\omega}^{4}\,,\,\,\underline{\omega}^{4}p=p\underline{\omega}^{4}-\lambda
t^{-1}\underline{\omega}^{3}.$

###  3.4 The smash product calculus

In all of this section, $B:=A^{\mathrm{co}H}\subseteq A$ is a trivial
extension for a right $H$-comodule algebra $A$. We denote the cleaving map by
$j\colon H\rightarrow A$; notice it is a right $H$-comodule algebra map with
convolution inverse $j^{-1}\colon H\rightarrow A$. We further employ the
$H$-action $\rhd\colon H\otimes B\rightarrow B$, defined by $h\rhd
b:=j(h_{1})bj^{-1}(h_{2})$ for $h\in H$ and $b\in B$. The properties of those
operations are discussed in Section 2.3 and we refer to them throughout the
section.

We now prove one of the main results of our paper, namely Theorem 1.1, which
amounts to the following. Given a right $H$-covariant FODC on $A$ we induce
FODC on $B$ and $H$ from the algebra homomorphisms $\iota\colon B\rightarrow
A$ and $j\colon H\rightarrow A$, respectively. Conversely, given FODC on $B$
and $H$, we can extend the algebra homomorphism $\theta\colon B\\#H\ni
b\\#h\mapsto bj(h)\in A$ to the corresponding morphism of right $H$-covariant
differential calculi, leading to a FODC $\Gamma_{\\#}$ on $A\cong B\\#H$ that
we shall call smash product calculus. This calculus is also considered in [47]
Theorem 2.7. If we apply these two procedures in sequence we reobtain the FODC
we started with.

######  Lemma 3.12.

Consider a FODC $(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and a right $H$-covariant
FODC $(\Gamma_{H},\mathrm{d}_{H})$ on $H$. On the $\Bbbk$-module

$\Gamma_{\\#}:=\Gamma_{B}\otimes H\oplus B\otimes\Gamma_{H}$ (47)

we have a left $A$-module structure by setting

$\begin{split}a\cdot(b\mathrm{d}_{B}b^{\prime}\otimes
h):=&a_{0}j^{-1}(a_{1})(a_{2}\rhd b)\mathrm{d}_{B}(a_{3}\rhd
b^{\prime})\otimes a_{4}h,\\\
a\cdot(b\otimes\omega^{H}):=&a_{0}j^{-1}(a_{1})(a_{2}\rhd b)\otimes
a_{4}\omega^{H},\end{split}$ (48)

for all $a\in A$, $b,b^{\prime}\in B$, $h\in H$ and $\omega^{H}\in\Gamma_{H}$.

###### Proof.

Let $a,a^{\prime}\in A$, $b,b^{\prime}\in B$ and $h\in H$. Then
$1\cdot(b\mathrm{d}_{B}b^{\prime}\otimes h)=b\mathrm{d}_{B}b^{\prime}\otimes
h$ and

$\displaystyle a\cdot(a^{\prime}\cdot(b\mathrm{d}_{B}b^{\prime}\otimes h))=$
$\displaystyle a\cdot(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})(a^{\prime}_{2}\rhd
b)\mathrm{d}_{B}(a^{\prime}_{3}\rhd b^{\prime}))$ $\displaystyle=$
$\displaystyle
a_{0}j^{-1}(a_{1})(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})(a^{\prime}_{2}\rhd
b)))\mathrm{d}_{B}(a_{3}\rhd(a^{\prime}_{3}\rhd b^{\prime}))$ $\displaystyle=$
$\displaystyle
a_{0}j^{-1}(a_{1})j(a_{2})a^{\prime}_{0}j^{-1}(a^{\prime}_{1})j^{-1}(a_{3})(a_{4}\rhd(a^{\prime}_{2}\rhd
b))\mathrm{d}_{B}(a_{5}\rhd(a^{\prime}_{3}\rhd b^{\prime}))$ $\displaystyle=$
$\displaystyle
a_{0}a^{\prime}_{0}j^{-1}(a_{1}a^{\prime}_{1})((a_{2}a^{\prime}_{2})\rhd
b)\mathrm{d}_{B}((a_{3}a^{\prime}_{3})\rhd b^{\prime})$ $\displaystyle=$
$\displaystyle(aa^{\prime})\cdot(b\mathrm{d}_{B}b^{\prime}\otimes h)$

prove that the first operation of (48) is a left $A$-action on
$\Gamma_{B}\otimes H$. In a similar fashion,
$1\cdot(b\otimes\omega^{H})=b\otimes\omega^{H}$ and

$\displaystyle a\cdot(a^{\prime}\cdot(b\otimes\omega^{H}))=$ $\displaystyle
a\cdot(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})(a^{\prime}_{2}\rhd b)\otimes
a^{\prime}_{3}\omega^{H})$ $\displaystyle=$ $\displaystyle
a_{0}j^{-1}(a_{1})(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})(a^{\prime}_{2}\rhd
b)))\otimes a_{3}a^{\prime}_{3}\omega^{H}$ $\displaystyle=$ $\displaystyle
a_{0}j^{-1}(a_{1})j(a_{2})a^{\prime}_{0}j^{-1}(a^{\prime}_{1})j^{-1}(a_{2})((a_{3}a^{\prime}_{2})\rhd
b)\otimes a_{4}a^{\prime}_{3}\omega^{H}$ $\displaystyle=$ $\displaystyle
a_{0}a^{\prime}_{0}j^{-1}(a_{1}a^{\prime}_{1})((a_{2}a^{\prime}_{2})\rhd
b)\otimes a_{3}a^{\prime}_{3}\omega^{H}$ $\displaystyle=$
$\displaystyle(aa^{\prime})\cdot(b\otimes\omega^{H})$

proves that the second operation of (48) is a left $A$-action on
$B\otimes\Gamma_{H}$. On $\Gamma_{\\#}$ this induces a left $A$-action by the
direct sum. ∎

######  Theorem 3.13.

Let $B:=A^{\mathrm{co}H}$ be a trivial Hopf-Galois extension.

  1. 1.

Any right $H$-covariant FODC $(\Gamma,\mathrm{d})$ induces a FODC
$(\Gamma_{B},\mathrm{d}_{B})$ on $B$ as the pullback calculus $\iota\colon
B\rightarrow A$ and a right $H$-covariant FODC $(\Gamma_{H},\mathrm{d}_{H})$
on $H$ as the pullback calculus $j\colon H\rightarrow A$;

  2. 2.

For any FODC $(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and right $H$-covariant FODC
$(\Gamma_{H},\mathrm{d}_{H})$ there is a right $H$-covariant FODC
$(\Gamma_{\\#},\mathrm{d}_{\\#})$ on $A$, where
$\Gamma_{\\#}=\Gamma_{B}\otimes H\oplus B\otimes\Gamma_{H}$ and

$\mathrm{d}_{\\#}\colon A\ni
a\mapsto\mathrm{d}_{\\#}a:=(\mathrm{d}_{B}(a_{0}j^{-1}(a_{1}))\otimes
a_{2}+a_{0}j^{-1}(a_{1})\otimes\mathrm{d}_{H}a_{2})\in\Gamma_{\\#}.$ (49)

The right $A$-action on $\Gamma_{\\#}$ is

$(a\cdot\mathrm{d}_{\\#}a^{\prime})\cdot
a^{\prime\prime}:=a\cdot(\mathrm{d}_{\\#}(a^{\prime}a^{\prime\prime}))-(aa^{\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime}$
(50)

for all $a,a^{\prime},a^{\prime\prime}\in A$;

  3. 3.

The above constructions 1. and 2. are inverse to each other in the sense that
their subsequent application produces calculi which are isomorphic to the
initial ones, as illustrated by the diagram

${(\Gamma,\mathrm{d})}$${(\Gamma_{B},\mathrm{d}_{B})~{}\&~{}(\Gamma_{H},\mathrm{d}_{H})}$${(\Gamma_{\\#},\mathrm{d}_{\\#})}$${(\Gamma_{B},\mathrm{d}_{B})~{}\text{\small{and}}~{}(\Gamma_{H},\mathrm{d}_{H})}$${(\Gamma_{\\#},\mathrm{d}_{\\#})}$${(\Gamma_{B}\otimes
1_{H},\mathrm{d}_{B}\otimes\mathrm{id}_{H})~{}\text{\small{and}}~{}(1_{B}\otimes\Gamma_{H},\mathrm{id}_{B}\otimes\mathrm{d}_{H})}$$\scriptstyle{\cong}$1.2.$\scriptstyle{\cong}$2.1.

where in the second line the first calculus on the left is isomorphic to the
first calculus on the right and the second calculus on the left is isomorphic
to the second calculus on the right;

###### Proof.

  1. 1.

This follows directly from Proposition 3.8. If $j\colon H\rightarrow A$ is not
injective we anyhow induce a FODC on $H$ according to Remark 3.5. This
calculus is right $H$-covariant because $j$ is right $H$-colinear.

  2. 2.

We first note that (49) is a well-defined $\Bbbk$-linear map
$\mathrm{d}_{\\#}\colon A\rightarrow\Gamma_{B}\otimes H\oplus
B\otimes\Gamma_{H}=\Gamma_{\\#}$. According to Lemma 3.12 the operations (48)
structure $\Gamma_{\\#}$ as a left $A$-module. We prove that
$\Gamma_{\\#}=A\cdot\mathrm{d}_{\\#}A$. Let $b\mathrm{d}_{B}b^{\prime}\otimes
h\in\Gamma_{B}\otimes H$ for $b,b^{\prime}\in B$ and $h\in H$. Then, by the
right $H$-colinearity of $j$, the fact that $j$ and $j^{-1}$ are convolution
inverse and that $\mathrm{d}_{B}$ satisfies the Leibniz rule,

$\displaystyle
b\cdot\mathrm{d}_{\\#}(b^{\prime}j(h))-(bb^{\prime})\cdot\mathrm{d}_{\\#}j(h)=$
$\displaystyle
b\cdot(\mathrm{d}_{B}(b^{\prime}j(h)_{0}j^{-1}(j(h)_{1}))\otimes
j(h)_{2}+b^{\prime}j(h)_{0}j^{-1}(j(h)_{1})\otimes\mathrm{d}_{H}(j(h)_{2}))$
$\displaystyle-(bb^{\prime})\cdot(\mathrm{d}_{B}(j(h)_{0}j^{-1}(j(h)_{1}))\otimes
j(h)_{2}+j(h)_{0}j^{-1}(j(h)_{1})\otimes\mathrm{d}_{H}(j(h)_{2}))$
$\displaystyle=$ $\displaystyle
b\cdot(\mathrm{d}_{B}(b^{\prime}j(h_{1})j^{-1}(h_{2}))\otimes
h_{3}+b^{\prime}j(h_{1})j^{-1}(h_{2})\otimes\mathrm{d}_{H}(h_{3}))$
$\displaystyle-(bb^{\prime})\cdot(\mathrm{d}_{B}(j(h_{1})j^{-1}(h_{2}))\otimes
h_{3}+j(h_{1})j^{-1}(h_{2})\otimes\mathrm{d}_{H}h_{3})$ $\displaystyle=$
$\displaystyle b\cdot(\mathrm{d}_{B}b^{\prime}\otimes
h+b^{\prime}\otimes\mathrm{d}_{H}h)-(bb^{\prime})\cdot(\mathrm{d}_{B}1\otimes
h+\otimes\mathrm{d}_{H}h)$ $\displaystyle=$ $\displaystyle
b\mathrm{d}_{B}b^{\prime}\otimes
h+bb^{\prime}\otimes\mathrm{d}_{H}h-bb^{\prime}\otimes\mathrm{d}_{H}h$
$\displaystyle=$ $\displaystyle b\mathrm{d}_{B}b^{\prime}\otimes h$

follows. In other words, we can recover any element in $\Gamma_{B}\otimes H$
as an element in $A\cdot\mathrm{d}_{\\#}A$. The same holds for
$B\otimes\Gamma_{H}$, namely let $b\otimes h\mathrm{d}_{H}h^{\prime}\in
B\otimes\Gamma_{H}$ for $b\in B$ and $h,h^{\prime}\in H$. Then

$\displaystyle(bj(h))\cdot\mathrm{d}_{\\#}(j(h^{\prime}))=$
$\displaystyle(bj(h))\cdot(\mathrm{d}_{B}(j(h^{\prime})_{0}j^{-1}(j(h^{\prime})_{1}))\otimes
j(h^{\prime})_{2}+j(h^{\prime})_{0}j^{-1}(j(h^{\prime})_{1})\otimes\mathrm{d}_{H}(j(h^{\prime})_{2}))$
$\displaystyle=$
$\displaystyle(bj(h))\cdot(\mathrm{d}_{B}(j(h^{\prime}_{1})j^{-1}(h^{\prime}_{2}))\otimes
h^{\prime}_{3}+j(h^{\prime}_{1})j^{-1}(h^{\prime}_{2})\otimes\mathrm{d}_{H}h^{\prime}_{3})$
$\displaystyle=$ $\displaystyle(bj(h))\cdot(1\otimes\mathrm{d}_{H}h^{\prime})$
$\displaystyle=$ $\displaystyle bj(h)_{0}j^{-1}(j(h)_{1})\otimes
j(h)_{2}\mathrm{d}_{H}h^{\prime}$ $\displaystyle=$ $\displaystyle
bj(h_{1})j^{-1}(h_{2})\otimes h_{3}\mathrm{d}_{H}h^{\prime}$ $\displaystyle=$
$\displaystyle b\otimes h\mathrm{d}_{H}(h^{\prime})$

follows in complete analogy. We conclude that
$\Gamma_{\\#}=A\cdot\mathrm{d}_{\\#}A$ is an equation of left $A$-modules.
Next, we verify that (50) is a right $A$-action on $\Gamma_{\\#}$. Let
$a,a^{\prime},a^{\prime\prime},a^{\prime\prime\prime}\in A$. In fact, we
obtain $(a\cdot\mathrm{d}_{\\#}a^{\prime})\cdot
1=a\cdot\mathrm{d}_{\\#}(a^{\prime}1)-(aa^{\prime})\cdot\mathrm{d}_{\\#}1=a\cdot\mathrm{d}_{\\#}a^{\prime}$
and

$\displaystyle((a\cdot\mathrm{d}_{\\#}a^{\prime})\cdot a^{\prime\prime})\cdot
a^{\prime\prime\prime}=$
$\displaystyle(a\cdot\mathrm{d}_{\\#}(a^{\prime}a^{\prime\prime})-(aa^{\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime})\cdot
a^{\prime\prime\prime}$ $\displaystyle=$ $\displaystyle
a\cdot\mathrm{d}_{\\#}(a^{\prime}a^{\prime\prime}a^{\prime\prime\prime})-(aa^{\prime}a^{\prime\prime})\cdot\mathrm{d}_{\\#}(a^{\prime\prime\prime})-(aa^{\prime})\cdot\mathrm{d}_{\\#}(a^{\prime\prime}a^{\prime\prime\prime})+(aa^{\prime}a^{\prime\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime\prime}$
$\displaystyle=$ $\displaystyle
a\cdot\mathrm{d}_{\\#}(a^{\prime}a^{\prime\prime}a^{\prime\prime\prime})-(aa^{\prime})\cdot\mathrm{d}_{\\#}(a^{\prime\prime}a^{\prime\prime\prime})$
$\displaystyle=$
$\displaystyle(a\cdot\mathrm{d}_{\\#}a^{\prime})\cdot(a^{\prime\prime}a^{\prime\prime\prime}).$

We further observe that the left and right $A$-actions on $\Gamma_{\\#}$
commute, i.e. for all $a,a^{\prime},a^{\prime\prime},a^{\prime\prime\prime}\in
A$ we have

$\displaystyle(a\cdot(a^{\prime}\cdot\mathrm{d}_{\\#}a^{\prime\prime}))\cdot
a^{\prime\prime\prime}=$
$\displaystyle((aa^{\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime})\cdot
a^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle(aa^{\prime})\cdot\mathrm{d}_{\\#}(a^{\prime\prime}a^{\prime\prime\prime})-(aa^{\prime}a^{\prime\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime\prime}$
$\displaystyle=$ $\displaystyle
a\cdot(a^{\prime}\cdot\mathrm{d}_{\\#}(a^{\prime\prime}a^{\prime\prime\prime})-(a^{\prime}a^{\prime\prime})\cdot\mathrm{d}_{\\#}a^{\prime\prime\prime})$
$\displaystyle=$ $\displaystyle
a\cdot((a^{\prime}\cdot\mathrm{d}_{\\#}a^{\prime\prime})\cdot
a^{\prime\prime\prime}).$

The Leibniz rule of $\mathrm{d}_{\\#}$ follows from (50) by setting $a=1$. It
remains to prove that the FODC $(\Gamma_{\\#},\mathrm{d}_{\\#})$ on $A$ is
right $H$-covariant. On $\Gamma_{\\#}=\Gamma_{B}\otimes H\oplus
B\otimes\Gamma_{H}$ we have the diagonal right $H$-coaction

$\displaystyle\Delta_{H}\colon\Gamma_{\\#}\ni\omega^{B}\otimes
h+b\otimes\omega^{H}\mapsto\omega^{B}\otimes h_{1}\otimes
h_{2}+b\otimes\omega^{H}_{0}\otimes\omega^{H}_{1}\in\Gamma_{\\#}\otimes H.$

It satisfies

$\displaystyle\Delta_{H}(a\cdot\mathrm{d}_{\\#}a^{\prime})=$
$\displaystyle\Delta_{H}(a\cdot(\mathrm{d}_{B}(a^{\prime}_{0}j^{-1}(a^{\prime}_{1}))\otimes
a^{\prime}_{2}+a^{\prime}_{0}j^{-1}(a^{\prime}_{1})\otimes\mathrm{d}_{H}a^{\prime}_{2}))$
$\displaystyle=$
$\displaystyle\Delta_{H}(a_{0}j^{-1}(a_{1})(a_{2}\rhd\mathrm{d}_{B}(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}a^{\prime}_{2}+a_{0}j^{-1}(a_{1})(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}\mathrm{d}_{H}a^{\prime}_{2})$ $\displaystyle=$
$\displaystyle\bigg{(}a_{0}j^{-1}(a_{1})(a_{2}\rhd\mathrm{d}_{B}(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}a^{\prime}_{2}+a_{0}j^{-1}(a_{1})(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}\mathrm{d}_{H}a^{\prime}_{2}\bigg{)}\otimes a_{4}a^{\prime}_{3}$
$\displaystyle=$
$\displaystyle\bigg{(}a_{0}\cdot(\mathrm{d}_{B}(a^{\prime}_{0}j^{-1}(a^{\prime}_{1}))\otimes
a^{\prime}_{2}+a^{\prime}_{0}j^{-1}(a^{\prime}_{1})\otimes\mathrm{d}_{H}a^{\prime}_{2})\bigg{)}\otimes
a_{1}a^{\prime}_{3}$ $\displaystyle=$
$\displaystyle(a_{0}\cdot\mathrm{d}_{\\#}a^{\prime}_{0})\otimes
a_{1}a^{\prime}_{1}$

for all $a,a^{\prime}\in A$, where we used that $(\Gamma_{H},\mathrm{d}_{H})$
is a right $H$-covariant FODC. Thus, $(\Gamma_{\\#},\mathrm{d}_{\\#})$ is
right $H$-covariant.

  3. 3.

Under the assumptions of 1., starting with a right $H$-covariant FODC
$(\Gamma,\mathrm{d})$ on $A$, we construct the FODC
$(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and a right $H$-covariant FODC
$(\Gamma_{H},\mathrm{d}_{H})$ on $H$ and then, via 2., the right $H$-covariant
smash product calculus $(\Gamma_{\\#},\mathrm{d}_{\\#})$ on $A$. We prove that
$(\Gamma,\mathrm{d})\cong(\Gamma_{\\#},\mathrm{d}_{\\#})$. The isomorphism is
given by $(\mathrm{id}_{A},\theta_{\Gamma})$, where

$\theta_{\Gamma}\colon\Gamma_{\\#}=\Gamma_{B}\otimes H\oplus
B\otimes\Gamma_{H}\ni b\mathrm{d}_{B}b^{\prime}\otimes
h+b^{\prime\prime}\otimes h^{\prime}\mathrm{d}_{H}h^{\prime\prime}\mapsto
b(\mathrm{d}b^{\prime})j(h)+b^{\prime\prime}j(h^{\prime})\mathrm{d}j(h^{\prime\prime})\in\Gamma$
(51)

is invertible with inverse

$\displaystyle\theta_{\Gamma}^{-1}\colon\Gamma\ni
a\mathrm{d}a^{\prime}\mapsto$ $\displaystyle(a_{0}j^{-1}(a_{1})\otimes
a_{2})\cdot(\mathrm{d}_{B}(a^{\prime}_{0}j^{-1}(a^{\prime}_{1}))\otimes
a^{\prime}_{2}+a^{\prime}_{0}j^{-1}(a^{\prime}_{1})\otimes\mathrm{d}_{H}a^{\prime}_{2})$
$\displaystyle=$ $\displaystyle
a_{0}j^{-1}(a_{1})\mathrm{d}_{B}(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}a^{\prime}_{2}+a_{0}j^{-1}(a_{1})(a_{2}\rhd(a^{\prime}_{0}j^{-1}(a^{\prime}_{1})))\otimes
a_{3}\mathrm{d}_{H}a^{\prime}_{2}\in\Gamma_{\\#}.$

The $A$-bilinearity and right $H$-colinearity of $\theta_{\Gamma}$ follows in
complete analogy to the arguments used in 2.

Under the assumptions of 2. we construct the right $H$-covariant FODC
$(\Gamma_{\\#},\mathrm{d}_{\\#})$ from a given FODC
$(\Gamma_{B},\mathrm{d}_{B})$ and a given right $H$-covariant FODC
$(\Gamma_{H},\mathrm{d}_{H})$ on $H$. From $\iota\colon B\rightarrow A$ we
obtain a FODC $(\widetilde{\Gamma}_{B},\widetilde{\mathrm{d}}_{B})$ via

$\displaystyle\widetilde{\Gamma}_{B}:=B\cdot\mathrm{d}_{\\#}B=B\cdot(\mathrm{d}_{B}B\otimes
1)=B\mathrm{d}_{B}B\otimes 1=\Gamma_{B}\otimes 1,$

where $\widetilde{\mathrm{d}}_{B}\colon B\ni
b\mapsto\mathrm{d}_{\\#}b=\mathrm{d}_{B}b\otimes 1$. Clearly
$\Phi_{B}\colon\Gamma_{B}\otimes 1\ni\omega^{B}\otimes
1\mapsto\omega^{B}\in\Gamma_{B}$ is a left $B$-module isomorphism. It is even
a $B$-bimodule isomorphism, since

$\displaystyle\Phi_{B}((b\mathrm{d}_{B}b^{\prime}\otimes 1)\cdot
b^{\prime\prime})=$
$\displaystyle\Phi_{B}((b\cdot\mathrm{d}_{\\#}(b^{\prime}\otimes 1))\cdot
b^{\prime\prime})$ $\displaystyle=$
$\displaystyle\Phi_{B}(b\cdot\mathrm{d}_{\\#}((b^{\prime}\otimes
1)(b^{\prime\prime}\otimes 1))-((b\otimes 1)(b^{\prime}\otimes
1))\cdot\mathrm{d}_{\\#}(b^{\prime\prime}\otimes 1))$ $\displaystyle=$
$\displaystyle\Phi_{B}(b\cdot\mathrm{d}_{\\#}(b^{\prime}b^{\prime\prime}\otimes
1)-(bb^{\prime})\cdot(\mathrm{d}_{B}b^{\prime\prime}\otimes 1))$
$\displaystyle=$
$\displaystyle\Phi_{B}(b\mathrm{d}_{B}(b^{\prime}b^{\prime\prime})\otimes
1-bb^{\prime}\mathrm{d}_{B}b^{\prime\prime}\otimes 1)$ $\displaystyle=$
$\displaystyle
b\mathrm{d}_{B}(b^{\prime}b^{\prime\prime})-bb^{\prime}\mathrm{d}_{B}b^{\prime\prime}$
$\displaystyle=$ $\displaystyle(b\mathrm{d}_{B}b^{\prime})b^{\prime\prime}$
$\displaystyle=$ $\displaystyle\Phi_{B}(b\mathrm{d}_{B}b^{\prime}\otimes
1)b^{\prime\prime}$

and satisfies $\Phi_{B}(\mathrm{d}_{\\#}(b\otimes
1))=\Phi_{B}(\mathrm{d}_{B}b\otimes
1)=\mathrm{d}_{B}b=\mathrm{d}_{B}(\Phi_{B}(b\otimes 1))$ for all
$b,b^{\prime},b^{\prime\prime}\in B$. This proves that
$(\Gamma_{B},\mathrm{d}_{B})$ and
$(\widetilde{\Gamma}_{B},\widetilde{\mathrm{d}}_{B})$ are isomorphic as FODC.
Similarly, $j\colon H\rightarrow A$ induces a pullback calculus
$(\widetilde{\Gamma}_{H},\widetilde{\mathrm{d}}_{H})$ on $H$, where

$\displaystyle\widetilde{\Gamma}_{H}:=j(H)\cdot\mathrm{d}_{\\#}(j(H))=\mathrm{span}_{\Bbbk}\\{1\otimes
h\mathrm{d}_{H}h^{\prime}~{}|~{}h,h^{\prime}\in H\\}=1\otimes\Gamma_{H}$

and $\widetilde{\mathrm{d}}_{H}\colon H\ni
h\mapsto\mathrm{d}_{\\#}(j(h))=1\otimes\mathrm{d}_{H}h$ using that

$\displaystyle j(h)\cdot\mathrm{d}_{\\#}(j(h^{\prime}))=$ $\displaystyle
j(h)\cdot(\mathrm{d}_{B}(j(h^{\prime})_{0}j^{-1}(j(h^{\prime})_{1}))\otimes
j(h^{\prime})_{2}+j(h^{\prime})_{0}j^{-1}(j(h^{\prime})_{1})\otimes\mathrm{d}_{H}(j(h^{\prime})_{2}))$
$\displaystyle=$ $\displaystyle j(h)\cdot(\mathrm{d}_{B}(1)\otimes
h^{\prime}+1\otimes\mathrm{d}_{H}h^{\prime})$ $\displaystyle=$ $\displaystyle
1\otimes h\mathrm{d}_{H}h^{\prime}$

for all $h,h^{\prime}\in H$. Then $\Phi_{H}\colon 1\otimes\Gamma_{H}\ni
1\otimes\omega^{H}\mapsto\omega^{H}\in\Gamma_{H}$ is a $\Bbbk$-module
isomorphism. It is $H$-bilinear since $h\cdot_{j}(1\otimes
h^{\prime}\mathrm{d}_{H}h^{\prime\prime})=j(h)\cdot(1\otimes
h^{\prime}\mathrm{d}_{H}h^{\prime\prime})=1\otimes
hh^{\prime}\mathrm{d}_{H}h^{\prime\prime}$ and

$\displaystyle(1\otimes h\mathrm{d}_{H}h^{\prime})\cdot_{j}h^{\prime\prime}=$
$\displaystyle(1\otimes h\mathrm{d}_{H}h^{\prime})\cdot j(h^{\prime\prime})$
$\displaystyle=$ $\displaystyle(j(h)\cdot\mathrm{d}_{\\#}(j(h^{\prime})))\cdot
j(h^{\prime\prime})$ $\displaystyle=$ $\displaystyle
j(h)\cdot\mathrm{d}_{\\#}(j(h^{\prime}h^{\prime\prime}))-j(hh^{\prime})\cdot\mathrm{d}_{\\#}(j(h^{\prime\prime}))$
$\displaystyle=$ $\displaystyle 1\otimes
h\mathrm{d}_{H}(h^{\prime}h^{\prime\prime})-1\otimes
hh^{\prime}\mathrm{d}_{H}h^{\prime\prime}$ $\displaystyle=$ $\displaystyle
1\otimes((h\mathrm{d}_{H}h^{\prime})h^{\prime\prime})$

by the Leibniz rule of $\mathrm{d}_{\\#}$ and since $j$ is an algebra map,
where $\cdot_{j}$ denotes the $H$-module action on $\widetilde{\Gamma}_{H}$
induced from $j$. $\Phi_{H}$ intertwines the differentials since

$\Phi_{H}(\widetilde{\mathrm{d}}_{H}(1\otimes
h))=\Phi_{H}(1\otimes\mathrm{d}_{H}h)=\mathrm{d}_{H}h=\mathrm{d}_{H}(\Phi_{H}(1\otimes
h))$

for all $h\in H$ and $\Phi_{H}$ is right $H$-colinear since

$\displaystyle\Phi_{H}(1\otimes
h\mathrm{d}_{H}h^{\prime})_{0}\otimes\Phi_{H}(1\otimes
h\mathrm{d}_{H}h^{\prime})_{1}=$
$\displaystyle(h\mathrm{d}_{H}h^{\prime})_{0}\otimes(h\mathrm{d}_{H}h^{\prime})_{1}$
$\displaystyle=$ $\displaystyle h_{1}\mathrm{d}_{H}h^{\prime}_{1}\otimes
h_{2}h^{\prime}_{2}$ $\displaystyle=$ $\displaystyle\Phi_{H}(1\otimes
h_{1}\mathrm{d}_{H}h^{\prime}_{1})\otimes h_{2}h^{\prime}_{2}$
$\displaystyle=$ $\displaystyle\Phi_{H}((1\otimes
h\mathrm{d}_{H}h^{\prime})_{0})\otimes(1\otimes
h\mathrm{d}_{H}h^{\prime})_{1}$

for all $h,h^{\prime}\in H$, where we used that $(\Gamma_{H},\mathrm{d}_{H})$
is right $H$-covariant. In conclusion,
$(\Gamma_{H},\mathrm{d}_{H})\cong(\widetilde{\Gamma}_{H},\widetilde{\mathrm{d}}_{H})$
as right $H$-covariant FODC.

This concludes the proof of the theorem. ∎

We summarize our result in a corollary.

######  Corollary 3.14.

Let $B:=A^{\mathrm{co}H}$ be a trivial Hopf-Galois extension. There is a one
to one correspondence between:

  1. $\bullet$

Right $H$-covariant FODC $(\Gamma,\mathrm{d})$ on $A$.

  2. $\bullet$

FODC $(\Gamma_{B},\mathrm{d}_{B})$ on $B$ and right $H$-covariant FODC
$(\Gamma_{H},\mathrm{d}_{H})$ on $H$.

We conclude this section with the definition and some remarks on horizontal
and base forms. Even though we are mainly interested in trivial Hopf-Galois
extensions $B=A^{\mathrm{co}H}\subseteq A$, the following definition and
observation hold for arbitrary comodule algebras $A$.

######  Definition 3.15.

Let $A$ be a right $H$-comodule algebra and $(\Gamma,\mathrm{d})$ a right
$H$-covariant FODC on $A$. We set $B:=A^{\mathrm{co}H}$ and call horizontal
forms the elements of the $B$-bimodule
$\Gamma^{\mathrm{hor}}:=A\mathrm{d}B\subseteq\Gamma$.

Note that there are different notions of horizontal forms in the literature,
e.g. [4], where $\Gamma_{BM}^{\mathrm{hor}}=A(\mathrm{d}B)A$ is considered as
an $A$-bimodule rather than a $B$-bimodule.

######  Observation 3.16.

The base forms $\Gamma_{B}=B\mathrm{d}B$ are characterized by the following
equation

$\Gamma_{B}=\Gamma^{\mathrm{hor}}\cap\Gamma^{\mathrm{co}H}$ (52)

of $B$-bimodules. Clearly $\Gamma_{B}\subseteq\Gamma^{\mathrm{co}H}$ and
$\Gamma_{B}\subseteq\Gamma^{\mathrm{hor}}$, i.e.
$\Gamma_{B}\subseteq\Gamma^{\mathrm{hor}}\cap\Gamma^{\mathrm{co}H}$. On the
other hand, any element $\omega\in\Gamma^{\mathrm{hor}}$ is of the form
$\omega=a^{i}\omega_{i}$ for some $a^{i}\in A$, where $\omega_{i}$ is a
basis333Note that $\Gamma_{B}$ is finitely generated projective as a left
$B$-module. of $\Gamma_{B}$. If
$\omega\in\Gamma^{\mathrm{hor}}\cap\Gamma^{\mathrm{co}H}$ it follows that

$a^{i}\omega_{i}\otimes 1=\omega\otimes
1=\Delta_{R}(\omega)=a^{i}_{0}\omega_{i}\otimes a^{i}_{1}.$

Since $\omega_{i}$ are linearly independent this implies $a^{i}\otimes
1=a^{i}_{0}\otimes a^{i}_{1}$ for every fixed index $i$. In other words
$a^{i}\in B$ and $\omega\in\Gamma_{B}$.

##  4 A sheaf-theoretic approach to noncommutative calculi

In this section we introduce covariant FODC on sheaves. Our main goal is prove
that a covariant FODC on the total space of a quantum principal bundle (QPB)
gives a FODC on the base sheaf and the structure Hopf algebra and vice versa.
Such an approach is necessary whenever the the base of the principal bundle
that we consider is not affine, hence there is no natural commutative algebra
representing it. Consequently it is necessary to take a different approach to
the theory of quantum principal bundles, namely to quantize them as quantum
spaces, that is spaces with a (noncommutative) sheaf of $H$-comodules. Then,
it is only natural to extend what we have proven in our previous section in
the affine setting to this more general framework and obtain differential
calculi on QPB over non-affine bases.

###  4.1 Differential calculus on sheaves

We briefly recall the key definition in [3] of quantum principal bundle and
then we proceed to give a quantum differential calculus on sheaves. The
previous sections can be understood as a local description of our sheaf-
theoretic construction.

A quantum ringed space $(M,\mathcal{O}_{M})$ is a pair consisting of a
classical topological space $M$ and a sheaf $\mathcal{O}_{M}$ on $M$ of
noncommutative algebras. A sheaf of $H$-comodule algebras $\mathcal{F}$ is an
$H$-locally cleft principal bundle or locally cleft quantum principal bundle
(QPB for short) over $(M,\mathcal{O}_{M})$ if there exists an open covering
$\\{U_{i}\\}$ of $M$ such that:

  1. i.)

$\mathcal{F}(U_{i})^{\mathrm{co}H}=\mathcal{O}_{M}(U_{i})$,

  2. ii.)

$\mathcal{F}$ is locally cleft, that is, $\mathcal{F}(U_{i})$ is a cleft
extension of $\mathcal{F}(U_{i})^{\mathrm{co}H}$.

The last property is equivalent to the existence of a collection of cleaving
maps $j_{i}\colon H\longrightarrow\mathcal{F}(U_{i})$. We remark that in [3]
we also take into account the slightly different notion of quantum locally
principal bundles, where locally we impose the condition of principal
$H$-comodule algebra as in [27]. We refer the reader to [3] Sec. 2 for a full
discussion on these definitions. We now give the definition of covariant FODC
on a sheaf of comodule algebras.

######  Definition 4.1.

Let $H$ be a Hopf algebra and $\mathcal{F}$ a sheaf of right $H$-comodule
algebras on a topological space $M$. A right $H$-covariant FODC on
$\mathcal{F}$ is a sheaf $\Upsilon$ of right $H$-covariant
$\mathcal{F}$-bimodules with a morphism

$\mathrm{d}\colon\mathcal{F}\longrightarrow\Upsilon$ (53)

of sheaves of right $H$-comodules, such that for a fixed open cover
$\\{U_{i}\\}$ of $M$ the following conditions are satisfied.

  1. i.)

$\mathrm{d}_{i}(fg)=(\mathrm{d}_{i}f)g+f\mathrm{d}_{i}g$, for all
$f,g\in\mathcal{F}(U_{i})$

  2. ii.)

$\Upsilon(U_{i})=\mathcal{F}(U_{i})\mathrm{d}_{i}\mathcal{F}(U_{i})$

where

$\mathrm{d}_{i}:=\mathrm{d}|_{U_{i}}\colon\mathcal{F}(U_{i})\longrightarrow\Upsilon(U_{i}).$
(54)

Similarly one defines left $H$-covariant and $H$-bicovariant FODC on sheaves
of left $H$-comodule algebras and $H$-bicomodule algebras, respectively. In
case the $H$-coaction is trivial we obtain a description of FODC on a sheaf of
algebras. Note that the local conditions i.) and ii.) in Definition 4.1 are
required to hold on a fixed open cover, not on all open covers of the
topological space. This reflects our definition of QPB, where the local
conditions have to be satisfied on a specified fixed open cover, as well. In
our main examples of semisimple algebraic groups modulo parabolic subgroups
this open cover is given by the data of a quantum section.

######  Example 4.2.

We give some examples of FODC on sheaves, starting with classical
constructions.

  1. i.)

The de Rham calculus $\Omega$ on the sheaf $\mathcal{F}_{G}$ of smooth
functions on a Lie group $G$. In fact we may view $G$ as a QPB on its quotient
$G/K$, $\pi:G\longrightarrow G/K$, where $K$ is a closed subgroup. Then for a
suitable (saturated) open cover $\\{V_{i}\\}_{i}$ ([54] Ch. 2), the QPB
$\mathcal{F}_{G}$ is given by
$\mathcal{F}_{G}(V_{i})=\mathcal{C}^{\infty}(V_{i})$, $V_{i}\cong K\times
U_{i}$ $U_{i}=\pi(V_{i})$. Locally
$\mathcal{F}_{G}|_{V_{i}}\cong\mathcal{F}_{K\times U_{i}}$. The pullback of
the left multiplication $K\times V_{i}\rightarrow V_{i}$ constitutes a right
$\mathcal{C}^{\infty}(K)$-coaction on $\mathcal{C}^{\infty}(V_{i})$,
structuring the latter as a right $H=\mathcal{C}^{\infty}(K)$-comodule
algebra444A suitable completion must be taken to write
$\mathcal{C}^{\infty}(K\times U_{i})$ as tensor product in order to have the
coaction of $H$ (see [10] for more details).. If $\Omega_{G}^{1}$,
$\Omega_{G/K}^{1}$ denote the differential forms on $G$ and $G/K$
respectively, we have that the usual de Rham differential

$d:\pi^{*}\mathcal{F}_{G}\longrightarrow\pi^{*}\Omega_{G}^{1}$

satisfies Def. 4.1 for $\mathcal{F}=\pi^{*}\mathcal{F}_{G}$ and
$\gamma=\pi^{*}\Omega_{G}^{1}$. The presence of $\pi^{*}$ is because we need
to consider our sheaves on open sets $U$ in $G/K$, so that $\pi^{-1}(U)$ are
saturated open sets in $G$ and allow for (right) actions of $K$ on
$\pi^{*}\mathcal{F}_{G}(U)$.

  2. ii.)

If $\\{U_{i}\\}_{i}$ consists of a single open set we retrieve the definition
of a covariant FODC as considered in this article as well as in [56] and [29].

  3. iii.)

Take $G$ as a simple complex algebraic group and $P$ a parabolic subgroup.
Then consider the sheaf of algebraic functions on $G$ and $P$. We can view the
principal bundle $\pi\colon G\longrightarrow G/P$, in the sheaf-theoretic
language, both classically and in the quantum setting as in [3]. In the
classical setting however, an example of FODC in the sheaf-theoretic language
is given by the sheaf of Kähler differentials on $G$. In this particular
setting we have as above $\mathcal{F}_{G}|_{V_{i}}\cong\mathcal{F}_{P\times
U_{i}}$, that is, there exist a local section for the map $\pi$; moreover
$V_{i}$ and $U_{i}$ can be chosen affine. This is a peculiar property of the
flag algebraic varieties $G/P$ and we are going to give a quantum version of
the Kähler differentials construction in the next section.

Another class of examples is discussed in the Section 4.4.

###  4.2 Ore extension of covariant calculi

We recall that if $a\in A$ is a (right) Ore element, we denote $A[a^{-1}]$ the
Ore localization of $A$ at the multiplicative right Ore set
$S=\\{a^{m}\\}_{m\in\mathbb{Z}_{\geq}0}$ (for more details see [36] Ch. 4,
Sec. 10 and [32] Sec. I.7.).

######  Lemma 4.3.

Let $(\Gamma,\mathrm{d})$ be a FODC on an associative unital algebra $A$. Let
$a$ be an Ore element in $A$, then
$\Gamma_{a}:=A[a^{-1}]\,\Gamma\,A[a^{-1}]:=A[a^{-1}]\otimes_{A}\Gamma\otimes_{A}A[a^{-1}]$,
together with the differential $\mathrm{d}_{a}$ defined as follows

$\mathrm{d}_{a}:A[a^{-1}]\longrightarrow\Gamma_{a},\qquad\mathrm{d}_{a}(f)=\begin{cases}\mathrm{d}f&f\in
A\\\ -a^{-1}\,\mathrm{d}a\,a^{-1}&f=a^{-1}\end{cases}$ (55)

and then extended to $A[a^{-1}]$ by the Leibniz rule, is a FODC on
$A[a^{-1}]$.

If $A$ is a right $H$-comodule algebra such that $\delta_{R}(a)\in A\otimes H$
is invertible and $(\Gamma,\mathrm{d})$ a right $H$-covariant FODC, the FODC
$(\Gamma_{a},\mathrm{d}_{a})$ is right $H$-covariant with respect to the right
$H$-coaction

$\Delta^{a}_{R}\colon\Gamma_{a}\ni f\mathrm{d}_{a}g\mapsto
f_{0}\mathrm{d}_{a}g_{0}\otimes f_{1}g_{1}\in\Gamma_{a}\otimes H,$ (56)

where $\delta^{a}_{R}\colon A[a^{-1}]\ni f\mapsto f_{0}\otimes f_{1}\in
A[a^{-1}]\otimes H$ is the extended right $H$-coaction.

###### Proof.

We need to verify the properties of Definition 3.1. The first two properties
are satisfied by definition. Moreover by the equality

$f\,\mathrm{d}g\,a^{-n}=f\,\mathrm{d}_{a}(ga^{-n})-f\,g\,\mathrm{d}_{a}(a^{-n}),$

where $f,g\in A$, we see that we can express any element of $\Gamma_{a}$ as an
element in $A[a^{-1}]\,\mathrm{d}_{a}A[a^{-1}]$.

Assume now that $A$ is a right $H$-comodule algebra and $\delta(a)\in A\otimes
H$ invertible. Then, we are able to extend the right $H$-coaction
$\delta_{R}\colon A\rightarrow A\otimes H$ on $A$ as an algebra homomorphism
to a $\Bbbk$-linear map $\delta^{a}_{R}\colon A[a^{-1}]\rightarrow
A[a^{-1}]\otimes H$, where we set
$\delta^{a}_{R}(a^{-1}):=(\delta_{R}(a))^{-1}$. We prove that in this case
(56) is well-defined. Let $\sum\nolimits_{i}f^{i}(\mathrm{d}g^{i})h^{i}=0$ for
$f^{i},h^{i}\in A[a^{-1}]$ and $g^{i}\in A$. Then, there exist unique
$f^{\prime i},h^{\prime i}\in A$ and $n_{i},m_{i}\in\mathbb{N}_{0}$ such that
$f^{i}=a^{-n_{i}}f^{\prime i}$, $h^{i}=h^{\prime i}a^{-m_{i}}$. Set
$n:=\mathrm{max}_{i}\\{n_{i},m_{i}\\}\in\mathbb{N}_{0}$. Then
$a^{n}\sum_{i}f^{i}(\mathrm{d}g^{i})h^{i}a^{n}\in\Gamma$ and thus

$\displaystyle 0=$
$\displaystyle\Delta_{R}(a^{n}\sum_{i}f^{i}(\mathrm{d}g^{i})h^{i}a^{n})$
$\displaystyle=$
$\displaystyle\sum_{i}\Delta_{R}(a^{n}f^{i}(\mathrm{d}g^{i})h^{i}a^{n})$
$\displaystyle=$
$\displaystyle\sum_{i}\delta_{R}(a^{n}f^{i})\cdot(\mathrm{d}g^{i}_{0}\otimes
g^{i}_{1})\cdot\delta_{R}(h^{i}a^{n})$ $\displaystyle=$
$\displaystyle\sum_{i}\delta_{R}(a^{n})\delta^{a}_{R}(f^{i})\cdot(\mathrm{d}g^{i}_{0}\otimes
g^{i}_{1})\cdot\delta^{a}_{R}(h^{i})\delta_{R}(a^{n}),$

since $\Gamma$ is a right $H$-covariant FODC on $A$. In the last equation we
used that $\delta^{a}_{R}$ is an algebra homomorphism. Since $\delta(a)$ is
invertible by assumption, so is $\delta(a^{n})=\delta(a)^{n}$. By multiplying
the above equation with $\delta(a)^{-n}$ from the left and from the right, we
obtain

$0=\sum_{i}\delta^{a}_{R}(f^{i})\cdot(\mathrm{d}g^{i}_{0}\otimes
g^{i}_{1})\cdot\delta^{a}_{R}(h^{i})=\Delta^{a}_{R}(\sum_{i}f^{i}(\mathrm{d}g^{i})h^{i}),$

which proves that $\Delta^{a}_{R}$ is well-defined. From Lemma 3.7 we conclude
that $(\Gamma_{a},\mathrm{d}_{a})$ is a right $H$-covariant FODC on
$A[a^{-1}]$. ∎

We call $(\Gamma_{a},\mathrm{d}_{a})$ the localization of
$(\Gamma,\mathrm{d})$ via the Ore element $a\in A$. Consider now a right
$H$-comodule algebra $A$ together with a right $H$-covariant FODC
$(\Gamma,\mathrm{d})$ on $A$ and Ore elements $a_{i}\in A$, $i=1,\ldots,n$.
Assume further that $j_{i}\colon H\rightarrow A_{i}$ are cleaving maps
corresponding to trivial extensions. From Lemma 4.3 it follows that there are
right $H$-covariant FODC $(\Gamma_{i},\mathrm{d}_{i})$ on $A_{i}$ and by
Theorem 3.13 each of them induces a right $H$-covariant FODC on $H$. The
natural question arises if the calculi on $H$ are related.

######  Proposition 4.4.

Let $(\Gamma,\mathrm{d})$ be a right $H$-covariant FODC on a right
$H$-comodule algebra $A$ and $a_{i}\in A$, $i=1,\ldots,n$, Ore elements such
that $\delta_{R}(a_{i})$ is invertible and $j_{i}\colon H\rightarrow A_{i}$
are cleaving maps corresponding to trivial extensions. Assume further that
there are right $H$-colinear algebra maps $\pi_{i}\colon A_{i}\rightarrow H$
such that $\pi_{i}\circ j_{i}=\mathrm{id}_{H}$. Then, the right $H$-covariant
FODC on $H$ induced by $j_{i}$ are isomorphic to each other, i.e.
$(\Gamma_{j_{i_{1}}},\mathrm{d}_{j_{i_{1}}})\cong(\Gamma_{j_{i_{2}}},\mathrm{d}_{j_{i_{2}}})$
as right $H$-covariant FODC for $1\leq i_{1},i_{2}\leq n$.

###### Proof.

It suffices to prove the proposition for two Ore elements $a,a^{\prime}\in A$.
By Lemma 4.3 there are right $H$-covariant FODC $(\Gamma_{a},\mathrm{d}_{a})$
on $A_{a}$ and $(\Gamma_{a^{\prime}},\mathrm{d}_{a^{\prime}})$ on
$A_{a^{\prime}}$. According to Theorem 3.13 they induce two right
$H$-covariant FODC on $H$, $(\Gamma^{a}_{H},\mathrm{d}^{a}_{H})$ and
$(\Gamma^{a^{\prime}}_{H},\mathrm{d}^{a^{\prime}}_{H})$, respectively. Recall
that
$\Gamma^{a}_{H}=H\cdot_{a}\mathrm{d}^{a}_{H}H:=j_{a}(H)\mathrm{d}_{a}j_{a}(H)\subseteq\Gamma_{a}$
and
$\Gamma^{a^{\prime}}_{H}=H\cdot_{a^{\prime}}\mathrm{d}^{a}_{H}H:=j_{a^{\prime}}(H)\mathrm{d}_{a^{\prime}}j_{a^{\prime}}(H)\subseteq\Gamma_{a^{\prime}}$.
There is an isomorphism

$\displaystyle\Phi\colon\Gamma^{a}_{H}\ni
j_{a}(h^{i})\mathrm{d}_{a}j_{a}(h_{i})\mapsto$
$\displaystyle(j_{a^{\prime}}\circ\pi_{a})(j_{a}(h^{i}))\mathrm{d}_{a^{\prime}}(j_{a^{\prime}}\circ\pi_{a})(j_{a}(h_{i}))$
$\displaystyle=$ $\displaystyle
j_{a^{\prime}}(h^{i})\mathrm{d}_{a^{\prime}}j_{a^{\prime}}(h_{i})\in\Gamma_{a^{\prime}}$

with inverse given by the same expression but $a$ and $a^{\prime}$
interchanged. It is immediate to check that $\Phi$ is $H$-bilinear. It is
right $H$-colinear all involved maps are. This implies
$(\Gamma^{a}_{H},\mathrm{d}^{a}_{H})\cong(\Gamma^{a^{\prime}}_{H},\mathrm{d}^{a^{\prime}}_{H})$.
∎

Let $A$ be a an associative unital algebra and $a_{i}\in A$, $i=1,\ldots,n$ be
Ore elements. For a multi-index
$I=(i_{1},\ldots,i_{n})\in\\{1,\ldots,n\\}^{n}$ we use the notation

$A_{I}:=A[a_{i_{1}}^{-1},\ldots,a_{i_{n}}^{-1}]:=(\cdots(A[a_{i_{1}}^{-1}])\cdots)[a_{i_{n}}^{-1}]$
(57)

for the $n$-fold localization. Given a FODC $(\Gamma,\mathrm{d})$ on $A$ we
set

$\Gamma_{I}:=A_{I}\Gamma A_{I}:=A_{I}\otimes_{A}\Gamma\otimes_{A}A_{I}$ (58)

and $\mathrm{d}_{I}:=(\cdots(\mathrm{d}_{a_{i_{1}}})\cdots)_{a_{i_{n}}}\colon
A_{I}\rightarrow\Gamma_{I}$.

######  Lemma 4.5.

Let $A$ be a an associative unital algebra and $a_{i}\in A$, $i=1,\ldots,n$ be
Ore elements such that subsequent localizations do not depend on the order.
Then, subsequent localizations of a FODC on $A$ are independent of the order,
as well.

###### Proof.

Consider a FODC $(\Gamma,\mathrm{d})$ on $A$. For two Ore elements
$a_{i},a_{j}\in A$ we have

$\displaystyle(\Gamma_{a_{i}})_{a_{j}}=$ $\displaystyle
A_{j}\otimes_{A}(A_{i}\otimes_{A}\Gamma\otimes_{A}A_{i})\otimes_{A}A_{j}$
$\displaystyle=$ $\displaystyle
A[a_{i}^{-1},a_{j}^{-1}]\otimes_{A}\Gamma\otimes_{A}A[a_{i}^{-1},a_{j}^{-1}]$
$\displaystyle=$ $\displaystyle
A_{i}\otimes_{A}(A_{j}\otimes_{A}\Gamma\otimes_{A}A_{j})\otimes_{A}A_{i}$
$\displaystyle=$ $\displaystyle(\Gamma_{a_{j}})_{a_{i}}$

as an equation of $A[a_{i},a_{j}]$-bimodules. In particular,

$(\mathrm{d}_{a_{i}})_{a_{j}},(\mathrm{d}_{a_{j}})_{a_{i}}\colon
A[a_{i}^{-1},a_{j}^{-1}]\rightarrow\Gamma_{(i,j)}$

have the same domain and codomain. Clearly, they coincide on $A$. By
definition (55) the same holds true for $a_{i}$ and $a_{j}$. Both
differentials are extended by the Leibniz rule, so
$(\mathrm{d}_{a_{i}})_{a_{j}}=(\mathrm{d}_{a_{j}})_{a_{i}}$. This concludes
the proof. ∎

###  4.3 Quantum differential calculi on QPB over projective varieties

Let $G$ be a complex semisimple algebraic group $P$ a parabolic subgroup, the
quotient $G/P$ is a projective variety and the projection $G\longrightarrow
G/P$ is a principal bundle. Let $\mathcal{O}_{q}(G)$ and
$\mathcal{O}_{q}(P)=\mathcal{O}_{q}(G)/I_{P}$ be Hopf algebras over
$\mathbb{C}_{q}:=\mathbb{C}[q,q^{-1}]$, quantizations of $\mathcal{O}(G)$ and
$\mathcal{O}(P)$, the coordinate algebras of $G$ and $P$, respectively. Once
we fix a projective embedding for $G/P$, obtained through the global sections
of an ample line bundle $\mathcal{L}$, we have the graded algebra
$\mathcal{O}(G/P)$.

We say that $s\in\mathcal{O}_{q}(G)$ is a quantum section if

$(\mathrm{id}\otimes\pi)\Delta(s)=s\otimes\pi(s)\;$

where $\pi:\mathcal{O}_{q}(G)\longrightarrow\mathcal{O}_{q}(P)$ and $\Delta$
is the coproduct of $\mathcal{O}_{q}(G)$.

Denote with $\\{s_{i}\\}_{i\in\mathcal{I}}$ a choice of linearly independent
elements in $\Delta(s)=\sum_{i\in\mathcal{I}}s^{i}\otimes s_{i}$, that
generate the homogeneous coordinate ring $\mathcal{O}_{q}(G/P)$. Assume
furthermore that $S_{i}=\\{s_{i}^{r},r\in\mathbb{Z}_{\geq 0}\\}$ is Ore,
$\mathcal{O}_{q}(G)S_{i}^{-1}$ the consequent Ore localization, and that
subsequent localizations do not depend on the order. In the specialization for
$q=1$, the $s_{i}$’s define an open cover of $G$, whose projection on $M=G/P$
gives an open cover $\\{U_{i}\\}$ of $M$ (see [3], Sec. 4 for more details).
Let us define the open sets

$U_{I}:=U_{i_{1}}\cap\dots\cap U_{i_{r}},\qquad I=(i_{1},\dots,i_{r}).$

We consider on $M$ the topology generated by the open sets $U_{I}$, which
hence form a base $\mathcal{B}$.

######  Observation 4.6.

Given a $\mathcal{B}$-sheaf $\mathcal{G}$, that is, an assignment
$U_{I}\longrightarrow\mathcal{G}(U_{I})$, for all $U_{I}\in\mathcal{B}$,
satisfying the sheaf conditions on $\mathcal{B}$, then $\mathcal{G}$ extends
uniquely to a sheaf on $M$ (see Prop. I-12 [16]). We shall abuse the notation
and denote the sheaf extending the $\mathcal{B}$-sheaf $\mathcal{G}$ with the
same letter $\mathcal{G}$. Similary, a map of $\mathcal{B}$-sheaves
$\mathcal{G}$ and $\mathcal{G}^{\prime}$, that is a collection of morphisms
$\mathcal{G}(U_{I})\longrightarrow\mathcal{G}^{\prime}(U_{I})$ commuting with
restrictions, extends uniquely to a morphism of the sheaves $\mathcal{G}$ and
$\mathcal{G}^{\prime}$ (again, see Prop. I-12 [16]).

The main result in [3], Sec. 4 states the following.

######  Theorem 4.7.

Let the notation be as above. Then

  1. i.)

the assignment

$U_{I}\mapsto\mathcal{O}_{M}(U_{I}):=k_{q}[s_{k_{1}}s_{i_{1}}^{-1},\dots,s_{k_{r}}s_{i_{r}}^{-1};1\leq
k_{j}\leq n]\subset\mathcal{O}_{q}(G)S_{i_{1}}^{-1}\dots S_{i_{r}}^{-1}$

defines a sheaf $\mathcal{O}_{M}$ on $M=G/P$;

  2. ii.)

the assignment

$U_{I}\mapsto\mathcal{F}_{G}(U_{I}):=\mathcal{O}_{q}(G)S_{i_{1}}^{-1}\dots
S_{i_{r}}^{-1},\ \qquad I=(i_{1},\dots,i_{r})$

defines a sheaf $\mathcal{F}_{G}$ of right $\mathcal{O}_{q}(P)$-comodule
algebras on the topological space $M$;

  3. iii.)

on the open cover
$\mathcal{F}_{G}^{\mathrm{co}\mathcal{O}_{q}(P)}(U_{i})=\mathcal{O}_{M}(U_{i})$
holds;

If the sheaf $\mathcal{F}_{G}$ is locally cleft then $\mathcal{F}_{G}$ is a
quantum principal bundle on the quantum ringed space $(M,\mathcal{O}_{M})$.

Since the sheaf $\mathcal{F}_{G}$ is obtained from the Hopf algebra
$\mathcal{O}_{q}(G)$, via the quantum section $s$, which determines
$\mathcal{O}_{q}(G/P)$, by an abuse of terminology, we say that
$\mathcal{O}_{q}(G)$ is a QPB over $\mathcal{O}_{q}(G/P)$. We also notice,
that, in the statement of Thm 4.7, we are implicitly using Obs. 4.6.

######  Observation 4.8.

We observe that, using the technique of Prop. 4.7 in [3], one can prove that
$\mathcal{F}_{G}^{\mathrm{co}\mathcal{O}_{q}(P)}(U_{I})=\mathcal{O}_{M}(U_{I})$,
for all $I$, thus extending the result (3) in Theorem 4.7. In particular,
since $\\{U_{I}\\}$ are a base for the topology on $M$, this gives the sheaf
equality $\mathcal{F}_{G}^{\mathrm{co}\mathcal{O}_{q}(P)}=\mathcal{O}_{M}$.

######  Theorem 4.9.

Let $(\Gamma,d)$ be a right covariant FODC on the Hopf algebra
$\mathcal{O}_{q}(G)$ and $\mathcal{F}_{G}$ as above. Then

  1. i.)

there is a right $\mathcal{O}_{q}(P)$-covariant FODC
$(\Upsilon_{G},\mathrm{d}_{G})$ on the sheaf $\mathcal{F}_{G}$, where

$\Upsilon_{G}(U_{I}):=\mathcal{F}_{G}(U_{I})\,\Gamma\,\mathcal{F}_{G}(U_{I}):=\mathcal{F}_{G}(U_{I})\otimes_{\mathcal{O}_{q}(G)}\Gamma\otimes_{\mathcal{O}_{q}(G)}\mathcal{F}_{G}(U_{I})$
(59)

$\mathrm{d}_{I}:=\mathrm{d}_{G}|_{U_{I}}\colon\mathcal{F}_{G}(U_{I})\rightarrow\Upsilon_{G}(U_{I})$,
$I=(i_{1},\ldots,i_{r})$, are the localizations of $(\Gamma,\mathrm{d})$;

  2. ii.)

the FODC $(\Upsilon_{G},\mathrm{d}_{G})$ on the sheaf $\mathcal{F}_{G}$
induces a FODC $(\Upsilon_{M},\mathrm{d}_{M})$ on the base sheaf
$\mathcal{O}_{q}(G/P)=\mathcal{O}_{M}$, where

$\Upsilon_{M}(U_{I}):=\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})\,(\mathrm{d}_{M})_{I}\,\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$
(60)

and
$(\mathrm{d}_{M})_{I}:=\mathrm{d}_{I}|_{\mathcal{F}^{\mathrm{co}H}(U_{I})}\colon\mathcal{F}^{\mathrm{co}H}(U_{I})\rightarrow\Upsilon_{M}(U_{I})$,
$I=(i_{1},\ldots,i_{r})$;

###### Proof.

By Lemma 4.3 and Lemma 4.5, $(\Upsilon_{G}(U_{I}),\mathrm{d}_{I})$ is a right
$\mathcal{O}_{q}(P)$-covariant FODC on $\mathcal{F}_{G}(U_{I})$. Using the
restriction morphisms
$r_{JI}\colon\mathcal{F}_{G}(U_{I})\rightarrow\mathcal{F}_{G}(U_{J})$ of the
sheaf $\mathcal{F}_{G}$ we define restrictions

$R_{JI}\colon\underbrace{\Upsilon_{G}(U_{I})}_{=\mathcal{F}_{G}(U_{I})\mathrm{d}_{I}\mathcal{F}_{G}(U_{I})}\ni
f\mathrm{d}_{I}g\mapsto
r_{JI}(f)\mathrm{d}_{J}(r_{JI}(g))\in\underbrace{\Upsilon_{G}(U_{J})}_{=\mathcal{F}_{G}(U_{J})\mathrm{d}_{J}\mathcal{F}_{G}(U_{J})}.$
(61)

Since all involved maps are right $\mathcal{O}_{q}(P)$-linear, so is $R_{JI}$,
and one easily verifies that

$R_{JI}(f\cdot(f^{\prime}\mathrm{d}_{I}g^{\prime})\cdot g)=r_{JI}(f)\cdot
R_{JI}(f^{\prime}\mathrm{d}_{I}g^{\prime})\cdot r_{JI}(g)$

for all $f,f^{\prime},g,g^{\prime}\in\mathcal{F}_{G}(U_{I})$, i.e. that
$R_{JI}$ is bilinear along $r_{JI}$. Clearly, $R_{KI}=R_{KJ}\circ R_{JI}$ and
$R_{II}=\mathrm{id}_{\Upsilon(U_{I})}$ by the presheaf properties of $r$, i.e.
$\Upsilon_{G}$ is a presheaf of right $\mathcal{O}_{q}(P)$-covariant
$\mathcal{F}_{G}$-bimodules. The sheaf property is checked on the base and it
is done as in Prop. 4.7 in [3].

In complete analogy, ii.) follows by Proposition 3.13. In both i.) and ii.) we
make use of Obs. 4.6. ∎

######  Theorem 4.10.

There is a one to one correspondence between:

  1. 1.

A right $\mathcal{O}_{q}(P)$-covariant FODC $(\Upsilon_{G},\mathrm{d}_{G})$ on
the sheaf $\mathcal{O}_{q}(G)$.

  2. 2.

A FODC $(\Upsilon_{M},\mathrm{d}_{M})$ on the sheaf $\mathcal{O}_{M}$ and a
right $\mathcal{O}_{q}(P)$-covariant FODC $(\Gamma_{P},\mathrm{d}_{P})$ on
$\mathcal{O}_{q}(P)$.

###### Proof.

By Theorem 3.13 we know that a right $\mathcal{O}_{q}(P)$-invariant FODC
$(\Upsilon_{G}(U_{i}),\mathrm{d}_{G}|_{U_{i}})$ on $\mathcal{F}_{G}(U_{i})$
induces a FODC $(\Upsilon_{M}(U_{i}),\mathrm{d}_{M}|_{U_{i}})$ on
$\mathcal{O}_{M}(U_{i})$ and a right $\mathcal{O}_{q}(P)$-covariant FODC
$(\Gamma_{P},\mathrm{d}_{P})$ on $\mathcal{O}_{q}(P)$ and conversely a FODC
$(\Upsilon_{M}(U_{i}),\mathrm{d}_{M}|_{U_{i}})$ on $\mathcal{O}_{M}(U_{i})$
and a right $\mathcal{O}_{q}(P)$-covariant FODC $(\Gamma_{P},\mathrm{d}_{P})$
on $\mathcal{O}_{q}(P)$ induce a right $\mathcal{O}_{q}(P)$-invariant FODC
$(\Upsilon_{G}(U_{i}),\mathrm{d}_{G}|_{U_{i}})$ on $\mathcal{F}_{G}(U_{i})$
via the smash product construction. Since calculi on $\mathcal{O}_{q}(P)$
induced by different Ore extensions are all isomorphic by Proposition 4.4 this
extends to $U_{I}$ by induction. ∎

We now turn to the sheaf of horizontal form and right $H$-coinvariant forms,
that we shall define in analogy to their local counterparts.

######  Definition 4.11.

Let the notation be as above. We define the presheaves of

  1. 1.

horizontal forms as $\Upsilon^{\mathrm{hor}}\colon
U_{I}\mapsto\Gamma^{\mathrm{hor}}_{I}:=\mathcal{F}_{G}(U_{I})\mathrm{d}_{I}\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$
with restriction morphisms

$R^{\mathrm{hor}}_{JI}\colon\Gamma_{I}^{\mathrm{hor}}\ni
f\mathrm{d}_{I}g\mapsto
r_{JI}(f)\mathrm{d}_{I}r^{\mathrm{co}H}_{JI}(g)\in\Gamma_{J}^{\mathrm{hor}};$
(62)

  2. 2.

right $H$-coinvariant forms as $\Upsilon^{\mathrm{co}H}\colon
U_{I}\mapsto\Gamma^{\mathrm{co}H}_{I}:=\\{\omega\in\Gamma_{I}~{}|~{}\Delta_{I}(\omega)=\omega\otimes
1\\}$ with restriction morphisms

$R^{\mathrm{co}H}_{JI}:=R_{JI}|_{\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})}\colon\Gamma^{\mathrm{co}H}_{I}\rightarrow\Gamma^{\mathrm{co}H}_{J};$
(63)

######  Proposition 4.12.

The presheaves $\Upsilon^{\mathrm{hor}}$ and $\Upsilon^{\mathrm{co}H}$ are
sheaves of right $H$-covariant $\mathcal{O}_{M}$-bimodules, respectively.

###### Proof.

For every open $U_{I}$ we have that $\Gamma^{\mathrm{hor}}_{I}$, is a right
$H$-covariant $\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$-bimodule (see after Def.
3.15). Via the restriction maps
$r_{JI}\colon\mathcal{F}_{G}(U_{I})\rightarrow\mathcal{F}_{G}(U_{J})$ of the
sheaf $\mathcal{F}_{G}$, corresponding to the inclusion $U_{J}\subseteq
U_{I}$, we define restriction maps

$r^{\mathrm{co}H}_{JI}:=r_{JI}|_{\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})}\colon\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})\rightarrow\mathcal{F}_{G}^{\mathrm{co}H}(U_{J})$
(64)

and

$R^{\mathrm{hor}}_{IJ}\colon\Gamma^{\mathrm{hor}}_{I}=\mathcal{F}_{G}(U_{I})\mathrm{d}_{I}\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})\ni
f\mathrm{d}_{I}g\mapsto
r_{JI}(f)\mathrm{d}_{J}r^{\mathrm{co}H}_{JI}(g)\in\mathcal{F}_{G}(U_{J})\mathrm{d}_{J}\mathcal{F}_{G}^{\mathrm{co}H}(U_{J})=\Gamma^{\mathrm{hor}}_{J}.$
(65)

Note that (64) is well-defined, since for all
$f\in\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$ we have

$\Delta_{R}^{J}(r^{\mathrm{co}H}_{JI}(f))=\Delta_{R}^{J}(r_{JI}(f))=r_{JI}(f_{0})\otimes
f_{1}=r_{JI}(f)\otimes 1=r^{\mathrm{co}H}_{JI}(f)\otimes 1$

by the right $H$-colinearity of $r_{JI}$. Moreover,
$r^{\mathrm{co}H}_{JI}(fg)=r_{JI}(fg)=r_{JI}(f)r_{JI}(f)=r^{\mathrm{co}H}_{JI}(f)r^{\mathrm{co}H}_{JI}(g)$
for all $f,g\in\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$, proving that
$r^{\mathrm{co}H}_{JI}$ is a morphism of right $H$-comodule algebras. Clearly
$r^{\mathrm{co}H}_{ii}=\mathrm{id}_{\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})}$
and $r^{\mathrm{co}H}_{jk}\circ r^{\mathrm{co}H}_{IJ}=r^{\mathrm{co}H}_{ik}$
for another open $U_{k}\subseteq U_{J}\subseteq U_{I}$. Thus,
$\mathcal{F}_{G}^{\mathrm{co}H}$ is a presheaf of right $H$-comodule algebras.

The restriction maps (65) are right $H$-colinear since

$\displaystyle\Delta_{R}^{J}(R^{\mathrm{hor}}_{JI}(f\mathrm{d}_{I}g))=$
$\displaystyle\Delta_{R}^{J}(r_{JI}(f)\mathrm{d}_{J}r^{\mathrm{co}H}_{JI}(g))$
$\displaystyle=$ $\displaystyle
r_{JI}(f)_{0}\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(g)_{0})\otimes
r_{JI}(f)_{1}r^{\mathrm{co}H}_{JI}(g)_{1}$ $\displaystyle=$ $\displaystyle
r_{JI}(f_{0})\mathrm{d}_{J}r^{\mathrm{co}H}_{JI}(g)\otimes f_{1}$
$\displaystyle=$ $\displaystyle
R^{\mathrm{hor}}_{JI}((f\mathrm{d}_{I}g)_{0})\otimes(f\mathrm{d}_{I}g)_{1}$

for all $f\in\mathcal{F}_{G}(U_{I})$ and
$g\in\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$ by the coinvariance of $g$ and the
right $H$-colinearity of $r_{JI}$ and $r^{\mathrm{co}H}_{JI}$. Moreover, for
all $f\in\mathcal{F}_{G}(U_{I})$ and
$g,h,k\in\mathcal{F}_{G}^{\mathrm{co}H}(U_{I})$ we obtain

$\displaystyle R^{\mathrm{hor}}_{JI}(h\cdot(f\mathrm{d}_{I}g)\cdot k)=$
$\displaystyle R^{\mathrm{hor}}_{JI}(hf(\mathrm{d}_{I}(gk)-g\mathrm{d}_{I}k))$
$\displaystyle=$ $\displaystyle
r_{JI}(hf)\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(gk))-r_{JI}(hfg)\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(k))$
$\displaystyle=$ $\displaystyle
r_{JI}(h)r_{JI}(f)\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(g)r^{\mathrm{co}H}_{JI}(k))-r_{JI}(h)r_{JI}(f)r_{JI}(g)\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(k))$
$\displaystyle=$ $\displaystyle
r_{JI}(h)r_{JI}(f)\mathrm{d}_{J}(r^{\mathrm{co}H}_{JI}(g))r^{\mathrm{co}H}_{JI}(k)$
$\displaystyle=$ $\displaystyle
r^{\mathrm{co}H}_{JI}(h)R^{\mathrm{hor}}_{JI}(f\mathrm{d}_{I}g)r^{\mathrm{co}H}_{JI}(k)$

i.e. $R^{\mathrm{hor}}_{JI}$ is bilinear along $r^{\mathrm{co}H}_{JI}$. Since
clearly $R^{\mathrm{hor}}_{II}=\mathrm{id}_{\Gamma^{\mathrm{hor}}_{I}}$ and
$R^{\mathrm{hor}}_{KJ}\circ r^{\mathrm{hor}}_{JI}=r^{\mathrm{hor}}_{KI}$, we
have that $\Upsilon^{\mathrm{hor}}$ is a presheaf of right $H$-covariant
$\mathcal{F}_{G}^{\mathrm{co}H}$-bimodules. From the $\mathcal{B}$-sheaf
properties of $\mathcal{F}_{G}$ and $\mathcal{F}_{G}^{\mathrm{co}H}$, we
obtain that $\Upsilon^{\mathrm{hor}}$ is a $\mathcal{B}$-sheaf, this time of
right $H$-covariant $\mathcal{F}_{G}^{\mathrm{co}H}$-bimodules. Then we obtain
a unique sheaf $\Upsilon^{\mathrm{hor}}$ by Obs 4.6. As for
$\Upsilon^{\mathrm{co}H}$, the proof goes along the same lines. ∎

Consider a bicovariant FODC $(\Gamma,\mathrm{d})$ on the Hopf algebra
$\mathcal{O}_{q}(G)$. By the previous theorem we have a right
$\mathcal{O}_{q}(P)$-covariant FODC $(\Upsilon,\mathrm{d})$ on the quantum
principal bundle $\mathcal{O}_{q}(G)$ and a FODC
$(\Upsilon_{B},\mathrm{d}_{B})$ on the sheaf $\mathcal{O}_{q}(G/P)$. Since by
assumption
$\mathcal{F}_{G}(U_{i})^{\mathrm{co}H}\subseteq\mathcal{F}_{G}(U_{i})$ are
trivial Hopf-Galois extensions, there are cleaving maps $j_{i}\colon
H\rightarrow\mathcal{F}_{G}(U_{i})$ with convolution inverses
$j_{i}^{-1}\colon H\rightarrow\mathcal{F}_{G}(U_{i})$. Both are right
$H$-colinear maps. There are left $H$-actions

$h\rhd_{i}b:=j_{i}(h_{1})bj^{-1}_{i}(h_{2})\in\mathcal{F}_{G}(U_{i})^{\mathrm{co}H}$
(66)

on $\mathcal{F}_{G}(U_{i})^{\mathrm{co}H}$, where $h\in H$ and
$b\in\mathcal{F}_{G}(U_{i})^{\mathrm{co}H}$. Then Theorem 3.16 implies the
following.

######  Theorem 4.13.

Consider a bicovariant FODC $(\Gamma,\mathrm{d})$ on the Hopf algebra
$\mathcal{O}_{q}(G)$. Then
$\Upsilon_{M}=\Upsilon^{\mathrm{co}H}\cap\Upsilon^{\mathrm{hor}}$ is an
equality of sheaves of $\mathcal{F}_{G}^{\mathrm{co}H}$-bimodules.

###### Proof.

The local statement, that is
$\Upsilon_{M}(U_{I})=\Upsilon^{\mathrm{co}H}(U_{I})\cap\Upsilon^{\mathrm{hor}}(U_{I})$
is the consequence of Obs. 3.16 and the fact that subsequent localizations,
beside not depending on the order, do not alter the locally cleft property. ∎

###  4.4 Example of a differential structure on sheaves

We now examine the example of FODC on the QPB $\mathcal{F}_{G}$ described
above, $\mathcal{O}_{q}(G)=\mathcal{O}_{q}(\mathrm{SL}_{n})$ over the quantum
projective space $\widetilde{\mathcal{O}_{q}}(\mathbf{P}^{n-1})$. Such QPB was
constructed in Sec. 5.2 in [3]; let us briefly recap the main points. We start
from the quantum algebra
$\mathcal{O}_{q}(\mathrm{SL}_{n}):=\mathcal{O}_{q}(\mathrm{M})[\det_{q}^{-1}]$,
where $\mathcal{O}_{q}(\mathrm{M})$ is the matrix bialgebra defined via the
Manin relations on the generators $a_{ij}$ (Def. 5.1, [3]).

We have that $a_{11}$ is a quantum section and since

$\Delta(a_{11})=\sum a_{1k}\otimes a_{k1}$

we can choose the $s_{i}$ of Thm 4.7 as $s_{i}=a_{i1}$ (see also [3] Sec. 5).

We then consider the multiplicative sets
$S_{i}:=\\{a_{i1}^{r}\,;\,r\in\mathbb{Z}_{\geq 0}\\}$. Such sets verify the
Ore condition and subsequent localizations do not depend on their order (Lemma
5.1, [3]). We can define a QPB $\mathcal{F}_{G}$ as follows:

$U_{I}\longmapsto\mathcal{F}_{G}(U_{I}):=\mathcal{O}_{q}(\mathrm{SL}_{n})S_{i_{1}}^{-1}\dots
S_{i_{s}}^{-1},\qquad I=\\{i_{1},\dots,i_{s}\\},$

$\mathcal{F}_{G}$ is a sheaf of $\mathcal{O}_{q}(P)$-comodule algebras, where
$P$ maximal parabolic in $\mathrm{SL}_{n}(\mathbb{C})$ and
$\mathcal{O}_{q}(P)=\mathcal{O}_{q}(\mathrm{SL}_{n})/(a_{i1},\,1<i\leq n)$.
Moreover $\mathcal{F}_{G}$ is a QPB on the quantum projective space
$\widetilde{\mathcal{O}_{q}}(\mathbf{P}^{n-1})$ (Sec. 5, [3]).

We now focus on the special case $n=2$, where we can make explicit calculation
and exhibit the FODC. Consider the principal bundle
$\wp:\mathrm{SL}_{2}(\mathbb{C})\longrightarrow\mathrm{SL}_{2}(\mathbb{C})/P\simeq\mathbf{P}^{1}(\mathbb{C})$,
where $P$ is again the upper Borel in $\mathrm{SL}_{2}(\mathbb{C})$. Take
$V_{i}$ to be the subset of matrices in $\mathrm{SL}_{2}(\mathbb{C})$ with
entry $(i,1)$ not equal to zero. We observe that $\\{V_{1},V_{2}\\}$ is an
open cover of $\mathrm{SL}_{2}(\mathbb{C})$. Define $U_{i}=\wp(V_{i})$ and
that $\\{U_{1},U_{2}\\}$ is an open cover of $\mathbf{P}^{1}(\mathbb{C})$
since $\wp$ is an open map. Indeed these are the cover obtained via the
quantum section $a_{11}$:

$\Delta(a_{11})=a_{11}\otimes a_{11}+a_{12}\otimes a_{21}$

In fact $V_{1}=\\{g\in\mathrm{SL}_{2}(\mathbb{C})\,|\,a_{11}(g)\neq 0\\}$ and
$V_{2}=\\{g\in\mathrm{SL}_{2}(\mathbb{C})\,|\,a_{21}(g)\neq 0\\}$.

Following the notation of Sec. 3.3.1, let us now denote:

$a_{11}=\alpha,\quad a_{12}=\beta\quad a_{21}=\gamma,\quad a_{22}=\delta$

The coaction in (45) uniquely extends to a coaction on these localizations by
$\delta_{R}\alpha^{-1}=\alpha^{-1}\otimes t^{-1}$,
$\delta\gamma^{-1}=\gamma^{-1}\otimes t^{-1}$. The coinvariant subalgebras for
$\mathcal{O}(\mathrm{SL}_{2})[\alpha^{-1}]$ and
$\mathcal{O}(\mathrm{SL}_{2})[\gamma^{-1}]$ respectively are isomorphic to
$\mathbb{C}[z]$ and $\mathbb{C}[w]$ with $z:=\alpha^{-1}\gamma$ and
$w:=\alpha\gamma^{-1}$ and they are the coordinate rings of the affine
algebraic varieties corresponding to the opens in
$\mathrm{SL}_{2}(\mathbb{C})/P\simeq\mathbf{P}^{1}(\mathbb{C})$.

The quantum deformation of this construction have been studied in [3] as an
example of quantum principle bundle over projective base. In analogy with the
classical case we then define $A_{1}:=A[\alpha^{-1}]$,
$A_{2}:=A[\gamma^{-1}]$, the non commutative localizations in the elements
$\alpha$ and $\gamma$ respectively, of the Hopf algebra
$A=\mathcal{O}_{q}(\mathrm{SL}_{2})$. The coinvariants are given by

$B_{1}=\mathbb{C}_{q}[\gamma\alpha^{-1}]\simeq\mathbb{C}_{q}[u],\qquad
B_{2}=\mathbb{C}_{q}[\alpha\gamma^{-1}]\simeq\mathbb{C}_{q}[v]~{}$

and the ringed space
$(\mathbf{P}^{1}(\mathbb{C}),\mathcal{O}_{\mathbf{P}^{1}(\mathbb{C})})$ can be
then easily constructed as follows

$\mathcal{O}_{\mathbf{P}^{1}(\mathbb{C})}(U_{i}):=B_{i}~{},~{}~{}\mathcal{O}_{\mathbf{P}^{1}(\mathbb{C})}(U_{12}):=B_{12}:=B_{1}[u^{-1}]~{},~{}~{}\mathcal{O}_{\mathbf{P}^{1}(\mathbb{C})}(\mathbf{P}^{1}(\mathbb{C})):=\mathbb{C}$

with the restriction map given by $r_{{12},2}:B_{\,2}\to B_{12}$, $v\mapsto
u^{-1}$ that is again well defined since on $U_{12}$ one has $uv=1=vu$.
Moreover one can observe that

$\mathcal{F}_{G}(U_{i}):=A_{i}~{},\quad\mathcal{F}_{G}(U_{12}):=A_{1}[\gamma^{-1}]=A_{2}[\alpha^{-1}]~{},\quad\mathcal{F}_{G}(\mathbf{P}^{1}(\mathbb{C}))=A~{},$

is a sheaf of $\mathcal{O}_{q}(P)$-comodule algebras on
$\mathbf{P}^{1}(\mathbb{C})$.

In the previous section we discussed how to induce an $H$ covariant
differential calculus on $A_{i}$, we denote by
$(\Gamma_{A_{i}},\mathrm{d}_{i})$ where
$\Gamma_{A_{i}}:=A_{i}\Gamma_{A}A_{i}$. Our next task is to show that, at
least in this case, $\Gamma_{A_{i}}$ is generated as a left $A_{i}$ module by
$\Gamma_{A}$. The main and unique difficulty is to give an explicit formula
for $\mathrm{d}\alpha^{-1}$ and $\mathrm{d}\gamma^{-1}$; the Leibniz rule
implies that
$\mathrm{d}\alpha^{-1}=-\alpha^{-1}\,(\mathrm{d}\alpha)\,\alpha^{-1}$ but
observe now that we don’t know, in principle, how to move $\alpha^{-1}$ to the
left of $\omega^{i}$. To fix this problem we use a dual maneuver inserting
$1=\alpha^{-1}\alpha$ to the left of $\omega^{i}$ and moving $\alpha$ to the
right by using (46). Explicitly we get

$(\mathrm{d}\alpha)\alpha^{-1}=\alpha^{-1}\alpha(\mathrm{d}\alpha)\alpha^{-1}=\left(\frac{1-q^{-1}}{\lambda}\right)\omega^{1}+\left(\frac{1-q}{\lambda}\right)\omega^{4}-\alpha^{-1}\beta\omega^{2}$

from which we get

$\mathrm{d}\alpha^{-1}=\left(\frac{q^{-1}-1}{\lambda}\right)\alpha^{-1}\omega^{1}+\left(\frac{q-1}{\lambda}\right)\alpha^{-1}\omega^{4}+\alpha^{-2}\beta\omega^{2}.$

Along the same line we also get

$\mathrm{d}\gamma^{-1}=\left(\frac{q^{-1}-1}{\lambda}\right)\gamma^{-1}\omega^{1}+\left(\frac{q-1}{\lambda}\right)\gamma^{-1}\omega^{4}+\gamma^{-2}\beta\omega^{2}.$

We can then conclude that every element $\rho\in\Gamma_{A_{i}}$ can be
uniquely written as $\sum_{k=1,..,4}\sigma_{k}\omega^{k}$ with $\sigma_{k}\in
A_{i}$.

We are now ready to discuss the bicovariant FODC
$(\Gamma_{B_{1}},\mathrm{d}_{1})$ that is left $B_{1}$ module generated by
$\mathrm{d}_{1}u=\mathrm{d}(\gamma\alpha^{-1})$, where explicitly we have

$\mathrm{d}_{1}u=-\alpha^{-2}\omega^{2}$

and left and right $B_{1}$ module action are related by the following equation

$(\mathrm{d}_{1}u)u=q^{2}u(\mathrm{d}_{1}u).$

In analogy if we focus on the algebra freely generated by the symbol
$v=\alpha\gamma^{-1}$ and we get the bicovariant FODC
$(\Gamma_{B_{2}},\mathrm{d}_{2})$ where

$\mathrm{d}_{2}v=q\gamma^{-2}\omega^{2}$

and

$(\mathrm{d}_{2}v)v=q^{-2}v(\mathrm{d}_{2}v).$

Note that as discussed before in the general case
$(\Gamma_{B_{i}},\mathrm{d}_{i})$ is a bicovariant first order differential
calculus.

Acknoledgements. The work of P.A. and T.W. is partially supported by INFN,
CSN4, Iniziativa Specifica GSS, and by Università del Piemonte Orientale. P.A.
is also affiliated to INdAM, GNFM (Istituto Nazionale di Alta Matematica,
Gruppo Nazionale di Fisica Matematica).

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