# On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

Giovanna Carnovale Juan Cuadra Dipartimento di Matematica Pura Universidad de
Almería ed Applicata Dpto. Álgebra y Análisis Matemático via Trieste 63
E-04120 Almería, Spain I-35121 Padua, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We introduce a family of three parameters $2$-dimensional algebras
representing elements in the Brauer group $BQ(k,H_{4})$ of Sweedler Hopf
algebra $H_{4}$ over a field $k$. They allow us to describe the mutual
intersection of the subgroups arising from a quasitriangular or
coquasitriangular structure. We also define a new subgroup of $BQ(k,H_{4})$
and construct an exact sequence relating it to the Brauer group of Nichols
$8$-dimensional Hopf algebra with respect to the quasitriangular structure
attached to the $2\times 2$-matrix with $1$ in the $(1,2)$-entry and zero
elsewhere.

MSC:16W30, 16K50

## Introduction

The Brauer group of a Hopf algebra is an extremely complicated invariant that
reflects many aspects of the Hopf algebra: its automorphisms group, its Hopf-
Galois theory, its second lazy cohomology group, (co)quasitriangularity, etc.
It is very difficult to describe all its elements and to find their
multiplication rules. For the most studied case, that of a commutative and
cocommutative Hopf algebra, these are the results known so far: the first
explicit computation was done by Long in [14] for the group algebra
$k{\mathbb{Z}}_{n},$ where $n$ is square-free and $k$ algebraically closed
with $char(k)\nmid n$; DeMeyer and Ford [12] computed it for
$k{\mathbb{Z}}_{2}$ with $k$ a commutative ring containing $2^{-1}$. Their
result was extended by Beattie and Caenepeel in [2] for $k{\mathbb{Z}}_{n},$
where $n$ is a power of an odd prime number and some mild assumptions on $k$.
In [4] Caenepeel achieved to compute the multiplication rules for a subgroup,
the so-called split part, of the Brauer group for a faithfully projective
commutative and cocommutative Hopf algebra $H$ over any commutative ring $k$.
These results were improved in [6] and allowed him to compute the Brauer group
of Tate-Oort algebras of prime rank. For a unified exposition of these results
the profuse monograph [5] is recommended.

Since the Brauer group was defined for any Hopf algebra with bijective
antipode ([7], [8]), it was a main goal to compute it for the smallest
noncommutative noncocommutative Hopf algebra: Sweedler’s four dimensional Hopf
algebra $H_{4}$, which is generated over the field $k$ ($char(k)\neq 2$) by
the group-like $g$, the $(g,1)$-primitive element $h$ and relations
$g^{2}=1,h^{2}=0,gh=-hg$. A first step was the calculation in [20] of the
subgroup $BM(k,H_{4},R_{0})$ induced by the quasitriangular structure
$R_{0}=2^{-1}(1\otimes 1+g\otimes 1+1\otimes g-g\otimes g).$ It was shown to
be isomorphic to the direct product of $(k,+)$, the additive group of $k$, and
$BW(k)$, the Brauer-Wall group of $k$. It was later proved in [9] that the
subgroups $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ arising from all the
quasitriangular structures $R_{t}$ and the coquasitriangular structures
$r_{s}$ of $H_{4}$ respectively, with $s,t\in k$, are all isomorphic.

In this paper we introduce a family of three parameters $2$-dimensional
algebras $C(a;t,s)$, for $a,t,s\in k,$ that represent elements in
$BQ(k,H_{4})$. They will allow us to shed a ray of light on the subgroup
structure of $BQ(k,H_{4})$ and will provide some evidences about the
difficulty of the computation of this group. The algebra $C(a;t,s)$ is
generated by $x$ with relation $x^{2}=a$ and has a $H_{4}$-Yetter-Drinfeld
module algebra structure with action and coaction:

$g\cdot x=-x,\quad h\cdot x=t,\quad\rho(x)=x\otimes g+s\otimes h.$

We list the main properties of these algebras in Section 2 (Lemma 2.1) and we
show that $C(a;t,s)$ is $H_{4}$-Azumaya if and only if $2a\neq st$. When
$s=lt$ they represent elements in $BM(k,H_{4},R_{l})$ and this subgroup is
indeed generated by the classes of $C(a;1,t)$ with $2a\neq t$ together with
$BW(k)$, Proposition 2.6. The same statement holds true for
$BC(k,H_{4},r_{l})$ when $t=sl$ replacing $C(a;1,t)$ by $C(a;s,1)$,
Proposition 2.5.

Using the description of $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ in terms
of these algebras, Section 3 is devoted to analyze the intersection of these
subgroups inside $BQ(k,H_{4})$. Let $i_{t}$ and $\iota_{s}$ denote the
inclusion map of the former and the latter respectively. It is known that
$BW(k)$ is contained in any of the above subgroups. Theorem 3.5 states that:

  1. (1)

$Im(i_{t})\cap Im(\iota_{s})\neq BW(k)$ iff $ts=1$. If this is the case,
$Im(i_{t})=Im(\iota_{s})$;

  2. (2)

$Im(i_{t})\cap Im(i_{s})\neq BW(k)$ if and only if $t=s$;

  3. (3)

$Im(\iota_{t})\cap Im(\iota_{s})\neq BW(k)$ if and only if $t=s$.

A remarkable property of our algebras is that they represent the same class in
$BQ(k,H_{4})$ if and only if they are isomorphic, Corollary 3.4.

A morphism from the automorphism group of $H_{4}$ to $BQ(k,H_{4})$ was
constructed in [19], allowing to consider $k^{\cdot 2}$ as a subgroup of
$BQ(k,H_{4})$. In Section 4 we show that the subgroup $BM(k,H_{4},R_{l})$ is
conjugated to $BM(k,H_{4},R_{l\alpha^{2}})$ inside $BQ(k,H_{4})$, for
$\alpha\in k^{\cdot}$, by a suitable representative of $k^{\cdot 2}$, Lemma
4.1.

Any $H_{4}$-Azumaya algebra possesses two natural ${\mathbb{Z}}_{2}$-gradings:
one stemming from the action of $g$ and one from the coaction (after
projection) of $g$. In Section 6 we introduce the subgroup
$BQ_{grad}(k,H_{4})$ consisting of those classes of $BQ(k,H_{4})$ that can be
represented by $H_{4}$-Azumaya algebras for which the two
${\mathbb{Z}}_{2}$-gradings coincide. On the other hand, the Drinfeld double
of $H_{4}$ admits a Hopf algebra map $T$ onto Nichols $8$-dimensional Hopf
algebra $E(2)$. This map is quasitriangular as $E(2)$ is equipped with the
quasitriangular structure $R_{N}$ corresponding to the $2\times 2$-matrix $N$
with $1$ in the $(1,2)$-entry and zero elsewhere, see (5.1). If we consider
the associated Brauer group $BM(k,E(2),R_{N})$, then Theorem 5.2 claims that
$T$ induces a group homomorphism $T^{*}$ fitting in the following exact
sequence

$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$

So in order to compute $BQ(k,H_{4})$ one should first understand
$BM(k,E(2),R_{N})$. This new problem cannot be attacked with the available
techniques for computations of groups of type BM, [20], [10], [11]. Those
computations were achieved by finding suitable invariants for a class by means
of a Skolem-Noether-like theory. In the Appendix we underline some obstacles
to the application of these techniques to the computation of
$BM(k,E(2),R_{N})$: the set of elements represented by algebras for which the
action of one of the standard nilpotent generators of $E(2)$ is inner
coincides with the set of classes represented by ${\mathbb{Z}}_{2}$-graded
central simple algebras and this is not a subgroup of $BM(k,E(2),R_{N}),$
Theorems 6.1, 6.3. Moreover, $BM(k,E(2),R_{N})$ seems to be much more complex
than the groups of type BM treated until now since, according to Proposition
5.3, each group $BM(k,H_{4},R_{t})$ may be viewed as a subgroup of it.

##  1 Preliminaries

In this paper $k$ is a field, $H$ will denote a Hopf algebra over $k$ with
bijective antipode $S$, coproduct $\Delta$ and counit $\varepsilon$. Tensor
products $\otimes$ will be over $k$ and, for vector spaces $V$ and $W$, the
usual flip map is denoted by $\tau:V\otimes W\to W\otimes V$. We shall adopt
the Sweedler-like notations $\Delta(h)=h_{(1)}\otimes h_{(2)}$ and
$\rho(m)=m_{(0)}\otimes m_{(1)}$ for coproducts and right comodule structures
respectively. For $H$ coquasitriangular (resp. quasitriangular), the set of
all coquasitriangular (resp. quasitriangular) structures will be denoted by
$\cal U$ (resp. $\cal T$).

Yetter-Drinfeld modules. Let us recall that if $A$ is a left $H$-module with
action $\cdot$ and a right $H$-comodule with coaction $\rho$ the two
structures combine to a left module structure for the Drinfeld double
$D(H)=H^{*,cop}\bowtie H$ of $H$ (cfr. [15]) if and only if they satisfy the
so-called Yetter-Drinfeld compatibility condition:

$\rho(l\cdot b)=l_{(2)}\cdot b_{(0)}\otimes
l_{(3)}b_{(1)}S^{-1}(l_{(1)}),\quad\forall l\in H,b\in A.$ (1.1)

Modules satisfying this condition are usually called Yetter-Drinfeld modules.
If $A$ is a left $H$-module algebra and a right $H^{op}$-comodule algebra
satisfying (1.1) we shall call it a Yetter-Drinfeld $H$-module algebra.

The Brauer group (see [7], [8]). Suppose that $A$ is a Yetter-Drinfeld
$H$-module algebra. The $H$-opposite algebra of $A$, denoted by
$\overline{A}$, is the underlying vector space of $A$ endowed with product
$a\circ c=c_{(0)}(c_{(1)}\cdot a)$ for every $a,c\in A$. The same action and
coaction of $H$ on $A$ turn $\overline{A}$ into a Yetter-Drinfeld $H$-module
algebra. Given two Yetter-Drinfeld $H$-module algebras $A$ and $B$ we can
construct a new Yetter-Drinfeld module $A\\#B$ whose underlying vector space
is $A\otimes B$, with action $h\cdot(a\otimes b)=h_{(1)}\cdot a\otimes
h_{(2)}\cdot b$ and with coaction $a\otimes b\mapsto a_{(0)}b_{(0)}\otimes
b_{(1)}a_{(1)}$. This object becomes a Yetter-Drinfeld module algebra if we
provide it with the multiplication

$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d.$

For every finite dimensional Yetter-Drinfeld module $M$ the algebras ${\rm
End}(M)$ and ${\rm End}(M)^{op}$ can be naturally provided of a Yetter-
Drinfeld module algebra structure through (1.2) and (1.3) below respectively:

$\begin{array}[]{l}(h\cdot f)(m)=h_{(1)}\cdot f(S(h_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)},\end{array}$ (1.2) $\begin{array}[]{l}(h\cdot
f)(m)=h_{(2)}\cdot f(S^{-1}(h_{(1)})\cdot m),\vspace{2pt}\\\
\rho(f)(m)=f(m_{(0)})_{(0)}\otimes f(m_{(0)})_{(1)}S(m_{(1)}),\end{array}$
(1.3)

where $h\in H,f\in End(M),m\in M.$ A finite dimensional Yetter-Drinfeld module
algebra $A$ is called $H$-Azumaya if the following module algebra maps are
isomorphisms:

$\begin{array}[]{ll}F\colon A\\#{\overline{A}}\rightarrow{\rm
End}(A),&F(a\\#b)(c)=ac_{(0)}(c_{(1)}\cdot b),\vspace{2pt}\\\
G\colon\overline{A}\\#{{A}}\rightarrow{\rm
End}(A)^{op},&G(a\\#b)(c)=a_{(0)}(a_{(1)}\cdot c)b.\end{array}$ (1.4)

The algebras ${\rm End}(M)$ and ${\rm End}(M)^{op}$, for a finite dimensional
Yetter-Drinfeld module $M$, provided with the preceding structures are
$H$-Azumaya.

The following relation $\sim$ established on the set of isomorphism classes of
$H$-Azumaya algebras is an equivalence relation: $A\sim B$ if there exist
finite dimensional Yetter-Drinfeld modules $M$ and $N$ such that $A\\#{\rm
End}(M)\cong B\\#{\rm End}(N)$ as Yetter-Drinfeld module algebras. The set of
equivalence classes of $H$-Azumaya algebras, denoted by $BQ(k,H)$, is a group
with product $[A][B]=[A\\#B]$, inverse element $[\overline{A}]$ and identity
element $[End(M)]$ for finite dimensional Yetter-Drinfeld modules $M$. This
group is called the full Brauer group of $H$. The adjective full is used to
distinguish it from the subgroups presented next, that receive the same name
in the literature.

Given a left $H$-module algebra $A$ with action $\cdot$ and a quasitriangular
structure $R=R^{(1)}\otimes R^{(2)}$ on $H$, a right $H^{op}$-comodule algebra
structure $\rho$ on $A$ is determined by

$\rho(a)=(R^{(2)}\cdot a)\otimes R^{(1)},\quad\forall a\in A.$

We will call this coaction the coaction induced by $\cdot$ and $R$. It is
well-known that $(A,\cdot,\rho)$ satisfies the Yetter-Drinfeld condition. This
allows the definition of the subgroup $BM(k,H,R)$ of $BQ(k,H)$ whose elements
are equivalence classes of $H$-Azumaya algebras with coaction induced by $R$
([8, §1.5]). To underline that a representative $A$ of a given class in
$BQ(k,H)$ represents a class in $BM(k,H,R)$ we shall say that $A$ is an
$(H,R)$-Azumaya algebra. The inclusion map will be denoted by $i\colon
BM(k,H,R)\to BQ(k,H)$. For $H$ finite dimensional $BQ(k,H)=BM(k,D(H),{\cal
R}),$ where ${\cal R}$ is the natural quasitriangular structure on the
Drinfeld double $D(H)$.

Dually, given a right $H^{op}$-comodule algebra $A$ with coaction $\varrho$
and a coquasitriangular structure $r$ on $H$, a $H$-module algebra structure
$\cdot$ on $A$ is determined by

$h\cdot a=a_{(0)}r(h\otimes a_{(1)}),\quad\forall a\in A,h\in H,$

and $(A,\cdot,\varrho)$ becomes a Yetter-Drinfeld module algebra. We will call
this action the action induced by $\chi$ and $r$. The subset $BC(k,H,r)$ of
$BQ(k,H)$ consisting of those classes admitting a representative whose action
is induced by $r$ is a subgroup ([8, §1.5]). To stress that a representative
$A$ of a class in $BQ(k,H)$ represents a class in $BC(k,H,r)$ we shall say
that $A$ is an $(H,r)$-Azumaya algebra. The inclusion of $BC(k,H,r)$ in
$BQ(k,H)$ will be denoted by $\iota\colon BC(k,H,r)\to BQ(k,H)$.

On Sweedler Hopf algebra. In the sequel we will assume that $char(k)\neq 2.$
Let $H_{4}$ be Sweedler Hopf algebra, that is, the Hopf algebra over $k$
generated by a grouplike element $g$ and an element $h$ with relations,
coproduct and antipode:

$g^{2}=1,\quad h^{2}=gh+hg=0,\quad\Delta(h)=1\otimes h+h\otimes g,\quad
S(g)=g,\quad S(h)=gh.$

The Hopf algebra $H_{4}$ has a family of quasitriangular (indeed triangular)
structures. They were classified in [18] and are given by:

$R_{t}=\frac{1}{2}(1\otimes 1+1\otimes g+g\otimes 1-g\otimes
g)+\frac{t}{2}(h\otimes h+h\otimes gh+gh\otimes gh-gh\otimes h),$

where $t\in k$. It is well-known that $H_{4}$ is self-dual so that $H_{4}$ is
also cotriangular. Let $\\{1^{*},g^{*},h^{*},(gh)^{*}\\}$ be the basis of
$H^{*}_{4}$ dual to $\\{1,g,h,gh\\}$. We will often make use of the Hopf
algebra isomorphism

$\begin{array}[]{rl}\phi\colon H_{4}&\to H_{4}^{*}\\\ 1&\mapsto
1^{*}+g^{*}=\varepsilon\\\ h&\mapsto h^{*}+(gh)^{*}\\\ g&\mapsto
1^{*}-g^{*}\\\ gh&\mapsto h^{*}-(gh)^{*}.\end{array}$

So, the cotriangular structures of $H_{4}$ can be obtained applying the
isomorphism $\phi\otimes\phi$ to the $R_{t}$’s. They are:

$\begin{array}[]{c|rrrr}r_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\ g&1&-1&0&0\\\
h&0&0&t&-t\\\ gh&0&0&t&t\\\ \end{array}$

The Drinfeld double $D(H_{4})=H_{4}^{*,cop}\bowtie H_{4}$ of $H_{4}$ is
isomorphic to the Hopf algebra generated by $\phi(h)\bowtie 1$,
$\phi(g)\bowtie 1$, $\varepsilon\bowtie g$ and $\varepsilon\bowtie h$ with
relations:

$\begin{array}[]{l}(\phi(h)\bowtie 1)^{2}=0;\\\ (\phi(g)\bowtie
1)^{2}=\varepsilon\bowtie 1;\\\ (\phi(h)\bowtie 1)(\phi(g)\bowtie
1)+(\phi(g)\bowtie 1)(\phi(h)\bowtie 1)=0;\\\ (\varepsilon\bowtie h)^{2}=0;\\\
(\varepsilon\bowtie h)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\varepsilon\bowtie h)=0;\\\ (\varepsilon\bowtie g)^{2}=\varepsilon\bowtie
1;\\\ (\phi(h)\bowtie 1)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\phi(h)\bowtie 1)=0;\\\ (\phi(g)\bowtie 1)(\varepsilon\bowtie
h)+(\varepsilon\bowtie h)(\phi(g)\bowtie 1)=0;\\\ (\varepsilon\bowtie
g)(\phi(g)\bowtie 1)=(\phi(g)\bowtie 1)(\varepsilon\bowtie g);\\\
(\phi(h)\bowtie 1)(\varepsilon\bowtie h)-(\varepsilon\bowtie h)(\phi(h)\bowtie
1)=(\phi(g)\bowtie 1)-(\varepsilon\bowtie g)\end{array}$

and with coproduct induced by the coproducts in $H_{4}$ and $H_{4}^{*,cop}$.
For $l\in H_{4}$ we will sometimes write $\phi(l)$ instead of $\phi(l)\bowtie
1$ and $l$ instead of $1\bowtie l$ for simplicity.

Let us recall that a Yetter-Drinfeld $H_{4}$-module $M$ with action $\cdot$
and coaction $\rho$ becomes a $D(H_{4})$-module by letting $1\bowtie l$ act as
$l$ for every $l\in H_{4}$ and $(\phi(l)\bowtie
1).m=m_{(0)}(\phi(l)(m_{(1)}))$ for $m\in M$. Conversely, a $D(H_{4})$-module
$M$ becomes naturally a Yetter-Drinfeld module with $H_{4}$-action obtained by
restriction and $H_{4}$-coaction given by

$\rho(m)=\frac{1}{2}(\phi(1+g).m\otimes 1+\phi(1-g).m\otimes
g+\phi(h+gh).m\otimes h+\phi(h-gh)\otimes gh).$

We will often switch from one notation to the other according to convenience.

Centers and centralizers. If $A$ is a Yetter-Drinfeld $H$-module algebra, and
$B$ is a Yetter-Drinfeld submodule algebra of $A$, the left and the right
centralizer of $B$ in $A$ are defined to be:

$C^{l}_{A}(B):=\\{a\in A~{}|~{}ba=a_{(0)}(a_{(1)}\cdot b)\ \forall b\in B\\},$
$C^{r}_{A}(B):=\\{a\in A~{}|~{}ab=b_{(0)}(b_{(1)}\cdot a)\ \forall b\in B\\}.$

For the particular case $B=A$ we have the right center $Z^{r}(A)$ and the left
center $Z^{l}(A)$ of $A$. Both are trivial when $A$ is $H$-Azumaya, [8,
Proposition 2.12].

##  2 Some low dimensional representatives in $BQ(k,H_{4})$

In this section we shall introduce a family of 2-dimensional representatives
of classes in $BQ(k,H_{4})$ that will turn out to be easy to compute with.
They appeared for the first time in [16] and a particular case of them is
treated in [1, Section 1.5].

Let $a,\,t,\,s\in k$. The algebra $C(a)$ generated by $x$ with relation
$x^{2}=a$ is acted upon by $H_{4}$ by

$g\cdot 1=1,\quad g\cdot x=-x,\quad h\cdot 1=0,\qquad h\cdot x=t,$

and it is a right $H_{4}$-comodule via

$\rho_{s}(1)=1\otimes 1,\quad\quad\rho_{s}(x)=x\otimes g+s\otimes h.$

It is not hard to check that $C(a)$ with this action and coaction is a left
$H_{4}$-module algebra and a right $H^{op}$-comodule algebra. We shall denote
it by $C(a;t,s)$.

###### Lemma 2.1

Let notation be as above.

  1. (1)

$C(a;t,s)$ is a Yetter-Drinfeld module algebra with the preceding structures.

  2. (2)

As a module algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if and
only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$ and
$t=\alpha t^{\prime}$.

  3. (3)

As a comodule algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if
and only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$
and $s=\alpha s^{\prime}$.

  4. (4)

As a Yetter-Drinfeld module algebra $C(a;t,s)\cong
C(a^{\prime};t^{\prime},s^{\prime})$ if and only if there exists $\alpha\in
k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$, $t=\alpha t^{\prime}$ and
$s=\alpha s^{\prime}$.

  5. (5)

The module structure on $C(a;t,s)$ is induced by its comodule structure and a
cotriangular structure $r_{l}$ if and only if $t=sl$.

  6. (6)

The comodule structure on $C(a;t,s)$ is induced by its module structure and a
triangular structure $R_{l}$ if and only if $s=lt$.

  7. (7)

The $H_{4}$-opposite algebra of $C(a;t,s)$ is $C(st-a;t,s)$.

  8. (8)

$C(a;t,s)$ is an $H_{4}$-Azumaya algebra if and only if $2a\neq st$.

Proof: Let $x$ and $y$ be algebra generators in $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively with $x^{2}=a$ and
$y^{2}=a^{\prime}$.

(1) We verify condition (1.1) for $b=x$ and $l=h$. The other cases are easier
to check.

$\begin{array}[]{l}h_{(2)}\cdot x_{(0)}\otimes
h_{(3)}x_{(1)}S^{-1}(h_{(1)})\\\ \hskip 48.36958pt=g\cdot x\otimes(-gh)+g\cdot
s\otimes(gh)(-gh)+h\cdot x\otimes g^{2}\\\ \hskip 56.9055pt+h\cdot s\otimes
gh+x\otimes hg+s\otimes h^{2}\\\ \hskip 48.36958pt=x\otimes gh+t\otimes
1-x\otimes gh\\\ \hskip 48.36958pt=\rho_{s}(h\cdot x).\end{array}$

(2) An algebra isomorphism $f\colon C(a;t,s)\to
C(a^{\prime};t^{\prime},s^{\prime})$ must map $x$ to $\alpha y$ for some
$\alpha\in k^{\cdot}$. Then $a=x^{2}=(\alpha y)^{2}=\alpha^{2}a^{\prime}$.
Besides, $h.f(x)=f(h.x)$ implies $t^{\prime}\alpha=t$. It is easy to verify
that the condition is also sufficient.

(3) In the above setup $\rho_{s^{\prime}}(f(x))=(f\otimes{\rm id})\rho_{s}(x)$
implies $s^{\prime}\alpha=s$. It is not hard to check that this condition is
also sufficient.

(4) It follows from the preceding statements.

(5) If the module structure on $C(a;t,s)$ is induced by its comodule structure
$\rho_{s}$ and some $r_{l}\in{\cal U},$ then $t=h\cdot x=xr_{l}(h\otimes
g)+sr_{l}(h\otimes h)=sl.$ Conversely, if $t=sl$, then

$\begin{array}[]{l}g\cdot 1=1=1r_{l}(g\otimes 1);\qquad h\cdot
1=0=1r_{l}(h\otimes 1);\vspace{2pt}\\\ g\cdot x=-x=xr_{l}(g\otimes
g)+sr_{l}(g\otimes h)=x_{(0)}r_{l}(g\otimes x_{(1)});\vspace{2pt}\\\ h\cdot
x=t=xr_{l}(h\otimes g)+sr_{l}(h\otimes h)=x_{(0)}r_{l}(h\otimes
x_{(1)}).\end{array}$

Therefore the action is induced by the coaction and $r_{l}$.

(6) If the comodule structure on $C(a;t,s)$ is induced by the action and some
$R_{l}\in{\cal T},$ then

$x\otimes g+s\otimes h=\rho_{s}(x)=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)}=\frac{1}{2}(2x\otimes g)+\frac{l}{2}(2t\otimes h)=x\otimes
g+lt\otimes h$

hence $s=lt$. Conversely, if $s=lt$ then

$\begin{array}[]{l}\rho_{s}(1)=1\otimes 1=(R_{l}^{(2)}\cdot 1)\otimes
R_{l}^{(1)},\\\ \rho_{s}(x)=x\otimes g+s\otimes h=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)},\end{array}$

so the comodule structure is induced by the action and $R_{l}$.

(7) $\overline{C(a;t,s)}$ has $1,\,x$ as a basis and $1$ is the unit. The
action and coaction on $1$ and $x$ are as for $C(a;t,s)$. By direct
computation, $x\circ x=x(g\cdot x)+s(h\cdot x)=-a+st,$ so
$\overline{C(a;t,s)}=C(st-a;t,s)$.

(8) The algebra $C(a;t,s)$ is $H_{4}$-Azumaya if and only if the maps $F$ and
$G$ defined in (1.4) are isomorphisms. The space $C(a;t,s)\\#C(a;t,s)$ has
ordered basis $1\\#1,\,1\\#x,\,x\\#1,\,x\\#x$ while ${\rm End}(C(a;t,s))$ has
basis $1^{*}\otimes 1,\,1^{*}\otimes x,\,x^{*}\otimes 1,\,x^{*}\otimes x$ with
the usual identification $C(a;t,s)^{*}\otimes C(a;t,s)\cong{\rm
End}(C(a;t,s))$. Then for every $b,c\in C(a;t,s)$ we have

$\begin{array}[]{l}F(b\\#c)(1)=bc,\quad F(b\\#c)(x)=bx(g\cdot c)+sb(h\cdot
c),\vspace{2pt}\\\ G(1\\#b)(c)=cb,\quad G(x\\#b)(c)=x(g\cdot c)b+s(h\cdot
c)b.\end{array}$

The matrices associated with $F$ and $G$ with respect to the given bases are
respectively

$\left(\begin{array}[]{cccc}1&0&0&a\\\ 0&1&1&0\\\ 0&st-a&a&0\\\
1&0&0&st-a\end{array}\right)\qquad\left(\begin{array}[]{cccc}1&0&0&a\\\
0&1&1&0\\\ 0&a&st-a&0\\\ 1&0&0&st-a\end{array}\right)$

whose determinants $-(st-2a)^{2}$ and $(st-2a)^{2}$ are nonzero if and only if
$2a\neq st$. $\Box$

We have seen so far that the algebras $C(a;s,t)$ can be viewed as
representatives of classes in $BM(k,H_{4},R_{l})$ or in $BC(k,H_{4},r_{l})$
for suitable $l\in k$. It is known that these groups are all isomorphic to
$(k,+)\times BW(k),$ where $BW(k)$ is the Brauer-Wall group of $k$. We aim to
find to which pair $(\beta,[A])\in(k,+)\times BW(k)$ do the class of
$C(a;t,s)$ correspond. The group $BM(k,H_{4},R_{0})$ was computed in [20]. The
computation of $BC(k,H_{4},r_{0})$ follows from self-duality of $H_{4}$. It
was shown in [9] that all groups $BC(k,H_{4},r_{t})$ (hence, dually, all
$BM(k,H_{4},R_{t})$) are isomorphic. We shall use the description of
$BM(k,E(1),R_{t})$ given in [11] beause this might allow generalizations. In
the mentioned paper the Brauer group $BM(k,E(n),R_{0})$ is computed for the
family of Hopf algebras $E(n)$, where $E(1)=H_{4}$. We shall recall first
where do the isomorphism of the different Brauer groups $BC$ and $BM$ stem
from. The notion of lazy cocycle plays a key role here.

We recall from [3] that a lazy cocycle on $H$ is a left 2-cocycle $\sigma$
such that twisting $H$ by $\sigma$ does not modify the product in $H$. In
other words: for every $h,l,m\in H,$

$\sigma(h_{(1)}\otimes l_{(1)})\sigma(h_{(2)}l_{(2)}\otimes
m)=\sigma(l_{(1)}\otimes m_{(1)})\sigma(h\otimes l_{(2)}m_{(2)})$ (2.1)
$\sigma(h_{(1)}\otimes
l_{(1)})h_{(2)}l_{(2)}=h_{(1)}l_{(1)}\sigma(h_{(2)}\otimes l_{(2)})$ (2.2)

It turns out that a lazy left cocycle is also a right cocycle. Given a lazy
cocycle $\sigma$ for $H$ and a $H^{op}$-comodule algebra $A$, we may construct
a new $H^{op}$-comodule algebra $A_{\sigma}$, which is equal to $A$ as a
$H^{op}$-comodule, but with product defined by:

$a\bullet b=a_{(0)}b_{(0)}\sigma(a_{(1)}\otimes b_{(1)}).$

The group of lazy cocycles for $H_{4}$ is computed in [3]. Lazy cocycles are
parametrized by elements $t\in k$ as follows:

$\begin{array}[]{c|rrrr}\sigma_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\
g&1&1&0&0\\\ h&0&0&\frac{t}{2}&\frac{t}{2}\\\
gh&0&0&\frac{t}{2}&-\frac{t}{2}\\\ \end{array}$

We have the following group isomorphisms:

  1. (2.3)

$\Psi_{t}:BC(k,H_{4},r_{0})\rightarrow
BC(k,H_{4},r_{t}),[A]\mapsto[A_{\sigma_{t}}],$ constructed in [9, Proposition
3.1].

  2. (2.4)

$\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow BC(k,H_{4},r_{t}),[A]\mapsto[A^{op}].$
We explain how $A^{op}$ is equipped with the corresponding structure. The left
$H_{4}$-module algebra $A$ becomes a right $H_{4}^{*}$-comodule algebra. Then
$A^{op}$ is a right $H_{4}^{*,op}$-comodule algebra. The quasitriangular
structure $R_{t}$ is a coquasitriangular structure in $H_{4}^{*}$. Then
$A^{op}$ may be endowed with the left $H_{4}^{*}$-action stemming from the
comodule structure and $R_{t}$. On the other hand, $A^{op}$ may be viewed as
an $H_{4}^{op}$-comodule algebra through the isomorphism
$\phi:H_{4}\rightarrow H_{4}^{*}$. The coquasitriangular structure $R_{t}$ on
$H_{4}^{*}$ corresponds to the coquasitriangular structure $r_{t}$ on $H_{4}$
via $\phi.$

An isomorphism between $BM(k,H_{4},R_{0})$ and $BM(k,H_{4},R_{t})$ can be
constructed combining the above ones. Thus, the crucial step is to analyze the
sought correspondence for $BM(k,H_{4},R_{0})$.

The Brauer group $BM(k,H_{4},R_{0})$ is computed in [20] through the split
exact sequence (see also [1, Theorem 3.8] for an alternative approach):

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k,+)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BM(k,H_{4},R_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{BW(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{1.}$

The map $j^{*}:BM(k,H_{4},R_{0})\to BW(k),[A]\mapsto[A]$ is obtained by
restricting the $H_{4}$-action of $A$ to a $k{\mathbb{Z}}_{2}$-action via the
inclusion map $j:k{\mathbb{Z}}_{2}\rightarrow H_{4}$. This map is split by
$\pi^{*}:BW(k)\rightarrow BM(k,H_{4},R_{0}),[B]\mapsto[B]$, where $B$ is
considered as an $H_{4}$-module by restriction of scalars via the algebra
projection $\pi:H_{4}\rightarrow k{\mathbb{Z}}_{2},g\mapsto g,h\mapsto 0$. A
class $[A]$ lying in the kernel of $j^{*}$ is a matrix algebra with an inner
action of $H_{4}$ such that the restriction to $k{\mathbb{Z}}_{2}$ is strongly
inner. Thus there exist uniquely determined $u,w\in A$ such that

$g\cdot a=uau^{-1},\quad h\cdot a=w(g\cdot a)-aw\quad\forall a\in A,$ (2.5)
$u^{2}=1,\quad wu+uw=0,\quad w^{2}=\beta,$ (2.6)

for certain $\beta\in k$. Mapping $[A]\mapsto\beta$ defines a group
isomorphism $\chi\colon Ker(j^{*})\\\ \cong(k,+)$. We will determine
$j^{*}([C(a;t,s)])$ and $\chi([C(a;t,s)]\pi^{*}j^{*}([C(a;t,s)]^{-1}))$
whenever this is well-defined. To this purpose, we will first describe all
products of two algebras of type $C(a;t,s)$.

###### Lemma 2.2

Let $x,y$ be generators for $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively, with relations,
$H_{4}$-actions and coactions as above. The product
$C(a;t,s)\\#C(a^{\prime};t^{\prime},s^{\prime})$ is isomorphic to the
generalized quaternion algebra with generators $X=x\\#1$ and $Y=1\\#y$,
relations, $H_{4}$-action and and $H_{4}$-coaction:

$X^{2}=a,\quad Y^{2}=a^{\prime},\quad XY+YX=st^{\prime},$ $g\cdot X=-X,\quad
g\cdot Y=-Y,\quad h\cdot X=t,\quad h\cdot Y=t^{\prime},$ $\rho(X)=X\otimes
g+s\otimes h,\quad\rho(Y)=Y\otimes g+s^{\prime}\otimes h.$

Proof: By direct computation:

$X^{2}=(x\\#1)(x\\#1)=a\\#1,\quad Y^{2}=(1\\#y)(1\\#y)=a^{\prime}\\#1,\quad
XY=x\\#y,$ $YX=(1\\#y)(x\\#1)=x\\#(g\cdot y)+s\\#(h\cdot
y)=-XY+st^{\prime}\\#1.$

The formulas for the action and the coaction follow immediately from the
definition of action and coaction on a $\\#$-product. $\Box$

Elements in $BW(k)$ are represented by graded tensor products of the following
three type of algebras: $C(1)$ generated by the odd element $x$ with
$x^{2}=1$; classically Azumaya algebras having trivial
${\mathbb{Z}}_{2}$-action; and $C(a)\\#C(1),$ where $C(a)$ is generated by the
odd element $y$ with $y^{2}=a\in k^{\cdot}$ ([13, Theorem IV.4.4]).

###### Proposition 2.3

For $a\neq 0$ let $[C(a;t,0)]\in BM(k,H_{4},R_{0})$ denote the class of
$C(a;t,0).$ Then

$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])\in(k,+)\times BW(k),$

so the group $BM(k,H_{4},R_{0})$ is generated by $BW(k)$ and the classes
$[C(a;1,0)]$.

Proof: It is clear that if $a\neq 0$ then $j^{*}([C(a;t,0)])=[C(a)]$ and that
$\pi^{*}([C(a)])=[C(a;0,0)]$. Thus, $[C(a;t,0)\\#C(-a;0,0)]\in{\rm
Ker}(j^{*})$. We shall compute its image through $\chi$. By Lemma 2.2,
$C(a;t,0)\\#C(-a;0,0)$ is generated by $X$ and $Y$ with relations,
$H_{4}$-action and $H_{4}$-coaction:

$X^{2}=a,\quad Y^{2}=-a,\quad XY+YX=0,$ $g\cdot X=-X,\quad g\cdot Y=-Y,\quad
h\cdot X=t,\quad h\cdot Y=0,$ $\rho(X)=X\otimes g,\quad\rho(Y)=Y\otimes g.$

We look for the element $w$ satisfying (2.5) and (2.6). This element must be
odd with respect to the ${\mathbb{Z}}_{2}$-grading induced by the $g$-action,
hence $w=\lambda X+\mu Y$ for some $\lambda,\mu\in k$. Condition $h\cdot
X=-wX-Xw$ implies $t=-2\lambda a$ and condition $h\cdot Y=-wY-Yw$ implies
$0=-2\mu a$ so $w^{2}=a\lambda^{2}=t^{2}(4a)^{-1}$. Thus
$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])$ and we have the first statement. For the
second one, let $(\beta,[A])\in(k,+)\times BW(k)$. If $\beta=0$ there is
nothing to prove. If $\beta\neq 0$, the class
$[C((4\beta)^{-1}t^{2};t,0)]=[C((4\beta)^{-1};1,0)]=(\beta,[C((4\beta)^{-1})])$,
so $BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ is generated by $BW(k)$ and the
$[C(a;1,0)]$ for $a\neq 0.$ $\Box$

###### Lemma 2.4

Let $A$ be a $D(H_{4})$-module algebra.

  1. (1)

If the $h$-action on $A$ is trivial, then $A$ is $(H_{4},R_{0})$-Azumaya if
and only if it is $(H_{4},R_{t})$-Azumaya for every $t\in k$.

  2. (2)

If the $\phi(h)$-action on $A$ is trivial, then $A$ is $(H_{4},r_{0})$-Azumaya
if and only if it is $(H_{4},r_{t})$-Azumaya for every $t\in k$.

  3. (3)

The representatives of $BW(k)$ in $BC(k,H_{4},r_{t})$ and $BM(k,H_{4},R_{s})$
all coincide when viewed inside $BQ(k,H_{4})$.

Proof: (1) It follows from the form of the elements in ${\cal T}$ that if $A$
is $(H_{4},R_{0})$-Azumaya and the action of $h$ on $A$ is trivial (i.e., if
it lies in $BW(k)$), then its comodule structure $\rho_{t}$ induced by $R_{t}$
coincides with the comodule structure $\rho_{0}$ induced by $R_{0}$. Hence,
the maps $F$ and $G$ with respect to the action and $\rho_{t}$ are the same as
the maps $F$ and $G$ with respect to the action and $\rho_{0}$, so $A$ is
$(H_{4},R_{t})$-Azumaya for every $t\in k$.

(2) It is proved as (1).

(3) The first statement shows that the representatives of $BW(k)$ inside the
different $BM(k,H_{4},R_{t})$ coincide. The second statement shows the same
for $BC(k,H_{4},r_{t})$. Therefore we may assume $s=t=0$. The elements of this
copy of $BW(k)$ consist of ${\mathbb{Z}}_{2}$-graded Azumaya algebras $A$
where the grading is induced by the action of $g$. The $h$-action is trivial.
If the coaction $\rho$ is induced by $R_{0}$, then $a\in A$ is odd if and only
if $\rho(a)=a\otimes g$. The action $\rightharpoonup$ induced on $A$ by
$r_{0}$ and $\rho$ is as follows: $h\rightharpoonup a=0$ for every $a\in A$
and $g\rightharpoonup a=-a$ if and only if $\rho(a)=a\otimes g$, that is, the
original action on $A$ and $\rightharpoonup$ coincide. Thus, the maps $F$ and
$G$ coincide in all cases and $A$ represents an element in $BW(k)\subset
BM(k,H_{4},R_{0})$ if and only if it represents an element in $BW(k)\subset
BC(k,H_{4},r_{0})$. $\Box$

###### Proposition 2.5

The group $BC(k,H_{4},r_{s})$ is generated by the Brauer-Wall group and the
classes $[C(a;s,1)]$ for $2a\neq s$.

Proof: We will first deal with the case $s=0$. We will show that the
isomorphism $\Phi_{0}:BM(k,H_{4},R_{0})\rightarrow BC(k,H_{4},r_{0}),$
$[A]\mapsto[A^{op}]$ in (2.4) maps $[C(a;1,0)]$ to $[C(a;0,1)]$ and
$BW(k)\subset BM(k,H_{4},R_{0})$ to $BW(k)\subset BC(k,H_{4},r_{0})$. The
class $[C(a;1,0)]$ is mapped to the class of the algebra $C(a)^{op}$ with
comodule structure

$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$

and $H_{4}$-action induced by the cotriangular structure $r_{0}$, that is,
$g\cdot x=-x$ and $h\cdot x=0$. The algebra $C(a)^{op}$ with these structures
is just $C(a;0,1)$.

Let $A$ be a representative of a class in $BW(k)\subset BM(k,H_{4},R_{0})$
with action $\cdot$ for which $h\cdot a=0$ for all $a\in A$. The class $[A]$
is mapped by $\Phi_{0}$ to the class of $A^{op}$ with coaction

$\rho(a)=a\otimes 1^{*}+(g\cdot a)\otimes g^{*}+(h\cdot a)\otimes
h^{*}+(gh\cdot a)\otimes(gh)^{*}\in A\otimes\phi(k{\mathbb{Z}}_{2}).$

Therefore $[A^{op}]\in BW(k)\subset BC(k,H_{4},r_{0})$.

We now take $s\in k$ arbitrary and use the isomorphism
$\Psi_{s}:BC(k,H_{4},r_{0})\rightarrow BC(k,H_{4},r_{s})$ in (2.3) to prove
the statement. We will show that $[C(a;0,1)]$ is mapped to
$[C(a+2^{-1}s;s,1)]$ through $\Psi_{s}$. Recall that $\Psi_{s}$ maps the class
of $C(a;0,1)$ to the class of the algebra $C(a;0,1)_{\sigma_{s}}$. It is
generated by $x$ with relation

$x\bullet x=x^{2}\sigma_{s}(g\otimes g)+x\sigma_{s}(h\otimes
g)+x\sigma_{s}(g\otimes h)+\sigma_{s}(h\otimes h)=a+\frac{s}{2},$

with (same) coaction $\rho(x)=x\otimes g+1\otimes h$ and action induced by
$\rho$ and $r_{s}$, that is:

$g\cdot x=r_{s}(g\otimes g)x+r_{s}(g\otimes h)=-x,\quad h\cdot
x=r_{s}(h\otimes g)x+r_{s}(h\otimes h)=s.$

Then $\Psi_{s}([C(a;0,1)])=[C(a+\frac{s}{2};s,1)]$.

Since the coaction is not changed by $\Psi_{s}$ the class of an element $A$
for which the image of the coaction is in $A\otimes k{\mathbb{Z}}_{2}$ is
again of this form. Hence the classes in $BW(k)\subset BC(k,H_{4},r_{0})$
correspond to the classes in $BW(k)\subset BC(k,H_{4},r_{s})$. $\Box$

###### Proposition 2.6

The group $BM(k,H_{4},R_{t})$ is generated by the Brauer-Wall group and the
classes $[C(a;1,t)]$ for $2a\neq t$.

Proof: Through the isomorphism $\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow
BC(k,H_{4},r_{t})$ in (2.4), the class $[C(a;1,t)]$ is mapped to $[C(a;t,1)]$
and the classes in $BW(k)\subset BM(k,H_{4},R_{t})$ correspond to the classes
in $BW(k)\subset BC(k,H_{4},r_{t})$. The $H_{4}$-comodule structure on the
algebra $C(a)^{op}$ is:

$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$

The $H_{4}$-action induced by the cotriangular structure $r_{t}$ on $H_{4}$
gives $h\cdot x=t$. Therefore this algebra is $C(a;t,1)$. Finally, the
statement concerning $BW(k)$ is proved as in the preceding theorem. $\Box$

###### Remark 2.7

That $BM(k,H_{4},R_{t})$ is generated by $BW(k)$ and the classes $[C(a;1,t)]$
for $2a\neq t$ was first discovered in [1, Theorem 3.8 and Page 392] as a
consequence of the Structure Theorems for $(H_{4},R_{t})$-Azumaya algebras.
Since we will strongly use Proposition 2.6 later, for the reader’s convenience
we offered this alternative and self-contained approach. Notice that it mainly
relies on Lemma 2.2 that will be another key result for us in the sequel.

##  3 Fitting $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ into $BQ(k,H_{4})$

As groups $BM(k,H_{4},R_{t})\cong BC(k,H_{4},r_{s})$ for every $s,t\in k$.
However, their images in $BQ(k,H_{4})$ through the natural embeddings

$i_{t}\colon BM(k,H_{4},R_{t})\to
BQ(k,H_{4})\quad\textrm{and}\quad\iota_{s}\colon BC(k,H_{4},r_{s})\to
BQ(k,H_{4})$

do not coincide in general. In this section we will describe the mutual
intersections of these images.

###### Proposition 3.1

Let $0\neq t\in k$ then $Im(i_{t})=Im(\iota_{t^{-1}})$

Proof: Given $t\neq 0$, by Lemma 2.1, $[C(a;1,t)]\in Im(i_{t})\cap
Im(\iota_{t^{-1}})$ for every $a\neq 2t.$ Besides, by Lemma 2.4,
$i_{t}(BW(k))=\iota_{s}(BW(k))$ for any $s\in k$. Since the elements of
$BW(k)$ and the $[C(a;1,t)]$’s generate $BM(k,H_{4},R_{t})$ and
$BC(k,H_{4},r_{t^{-1}})$ we are done. $\Box$

Given $[A]$ in $BQ(k,H_{4})$, there are two natural
${\mathbb{Z}}_{2}$-gradings on $A$, the one coming from the $g$-action, for
which $|a|=1$ iff $g\cdot a=-a$ for $0\neq a\in A$ and the one arising from
the coaction, for which $\deg(a)=1$ if and only if $({\rm
id}\otimes\pi)\rho(a)=a\otimes g$ where $\pi$ is the projection onto
$k{\mathbb{Z}}_{2}$. If we view $A$ as a $D(H_{4})$-module, the grading
$|\cdot|$ is associated with the $1\bowtie g$-action whereas the grading
$\deg$ is associated with the $\phi(g)\bowtie 1$-action. Let us observe that
for the classes $C(a;t,s)$ the two natural gradings coincide, for every
$a,\,t,\,s\in k$.

###### Lemma 3.2

Let $[A]\in BQ(k,H_{4})$ and $[B]$ in $i_{0}(BW(k))$. As a $H_{4}$-module
algebra,

  * (1)

$A\\#B\cong A\widehat{\otimes}B$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $\deg$-grading on $A$ and the natural $|\cdot|$-grading on
$B$.

  * (2)

$B\\#A\cong B\widehat{\otimes}A$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $|\cdot|$-grading on $A$ and the natural $|\cdot|$-grading
on $B$.

Proof: The two gradings on $B$ coincide and we have, for homogeneous $b\in B$
and $c\in A$ (for the $\deg$-grading):

$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d=ac\\#(g^{\deg(c)}\cdot
b)d=(-1)^{\deg(c)|b|}ac\\#bd.$

For homogeneous $b\in B$ and $c\in A$ (for the $|\cdot|$-grading):

$(d\\#c)(b\\#a)=db_{(0)}\\#(b_{(1)}\cdot c)a=db\\#(g^{|b|}\cdot
c)a=(-1)^{|c||b|}db\\#ca.$

$\Box$

It follows from Propositions 2.5, 2.6 and Lemma 3.2 that all elements in
$Im(i_{t})$ and $Im(\iota_{t})$ can be represented by algebras for which the
two ${\mathbb{Z}}_{2}$-gradings coincide, since this property is respected by
the $\\#$-product. Indeed, this kind of representatives give rise to a
subgroup that we will study in Section 5.

We will show now that groups of type $BC$ or $BM$ either intersect only in
$BW(k)$ or coincide and that the latter happens only in the situation of
Proposition 3.1.

###### Theorem 3.3

Consider the class of $C(a;t,s)$ in $BQ(k,H_{4})$. Then:

  1. (1)

$[C(a;t,s)]\in Im(i_{l})$ if and only if $s=lt$;

  2. (2)

$[C(a;t,s)]\in Im(\iota_{l})$ if and only if $sl=t$.

Proof: (1) We know from Lemma 2.1 that if the action (resp. coaction) of
$C(a;t,s)$ comes from the cotriangular (resp. triangular) structure, then the
indicated relations among the parameters hold. We only need to show that the
condition is still necessary if we change representative in the class.

Let us assume that $[C(a;t,s)]\in Im(i_{l})$ for some $l\in k$. Then
$[C(a;t,s)]=[C(b;1,l)][A]=[A][C(b;1,l)]$ for some $[A]\in i_{l}(BW(k))$ and
$b\in k$ with $2b\neq l$. Hence $[C(a;t,s)\\#C(l-b;1,l)]=[A]\in i_{l}(BW(k))$.
We may choose $A$ so that the $h$-action and the $\phi(h)$-action on $A$ are
trivial.

Since $[C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}]$ is trivial in $BQ(k,H_{4})$,
there is a $D(H_{4})$-module $P$ such that
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}\cong{\rm End(P)}$ as $D(H_{4})$-module
algebras. Then ${\rm End(P)}$ has a strongly inner $D(H_{4})$-action. In other
words, there is a convolution invertible algebra map $\nu\colon
D(H_{4})\to{\rm End}(P)$ such that

$(m\bowtie n)\cdot f=\nu(m_{(2)}\bowtie n_{(1)})f\nu^{-1}(m_{(1)}\bowtie
n_{(2)})$

for every $m\bowtie n\in D(H_{4}),f\in{\rm End(P)}$, where $\nu^{-1}$ denotes
the convolution inverse of $\nu$. In particular, for $u=\nu(\varepsilon\bowtie
g)$ and $w=\nu(\varepsilon\bowtie h)u$ we have

$\begin{array}[]{l}g\cdot f=ufu^{-1},\quad h\cdot f=w(g\cdot
f)-fw,\vspace{5pt}\\\ u^{2}=1,\quad w^{2}=0,\quad uw+wu=0.\end{array}$

We should be able to find $U,\,W\in C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that

$\begin{array}[]{l}U^{2}=1,\quad g\cdot Z=UZU^{-1},\vspace{5pt}\\\ g\cdot
W=-W,\quad W^{2}=0,\quad h\cdot Z=W(g\cdot Z)-ZW\end{array}$

for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$

Using the presentation of $C(a;t,s)\\#C(l-b;1,l)$ in Lemma 2.2 we may write
$W=\sum_{0\leq i,j\leq 1}X^{i}Y^{j}\\#\alpha_{ij}$ with
$\alpha_{ij}\in\overline{A}$ homogeneous of degree $i+j+1\;{\rm mod}\ 2$ with
respect to the $g$-grading. Since the action of $h$ on $1\\#\overline{A}$ is
trivial we have, for homogeneous $\gamma\in\overline{A}$:

$\begin{array}[]{ll}0&=h\cdot(1\\#\gamma)\vspace{2pt}\\\
&=W(g\cdot(1\\#\gamma))-(1\\#\gamma)W\vspace{2pt}\\\
&=(-1)^{|\gamma|}\sum_{0\leq i,j\leq
1}X^{i}Y^{j}\\#\alpha_{ij}\gamma-\sum_{0\leq i,j\leq
1}(X^{i}Y^{j})_{(0)}\\#((X^{i}Y^{j})_{(1)}\cdot\gamma)\alpha_{ij}\vspace{2pt}\\\
&=(-1)^{|\gamma|}[1\\#\alpha_{00}\gamma+Y\\#\alpha_{01}\gamma+X\\#\alpha_{10}\gamma+XY\\#\alpha_{11}\gamma]\vspace{2pt}\\\
&\phantom{=}-1\\#\gamma\alpha_{00}-Y\\#(-1)^{|\gamma|}\gamma\alpha_{01}-X\\#(-1)^{|\gamma|}\gamma\alpha_{10}-XY\\#\gamma\alpha_{11}.\end{array}$

From here we deduce that the odd elements $\alpha_{00},\alpha_{11}$ and the
even elements $\alpha_{10},\alpha_{01}$ belong to the
${\mathbb{Z}}_{2}$-center of $\overline{A}.$ Hence $\alpha_{00},\alpha_{11}$
are zero and $\alpha_{10},\alpha_{01}$ are scalars. So, we can write $W=\alpha
X\\#1+\beta Y\\#1$ for some $\alpha,\beta\in k$ and we will get:

$\begin{array}[]{rl}\alpha t+\beta&=h\cdot W=-2W^{2}=0,\\\
t&=h\cdot(X\\#1)=\alpha(-2a+ts),\\\ 1&=h\cdot(Y\\#1)=-\alpha
s-2\beta(l-b)=\alpha(-s+2t(l-b)).\end{array}$

Combining the second equation with the third one multiplied by $t$ and using
$\alpha\neq 0$ we obtain

$a=ts-t^{2}(l-b).$ (3.1)

The $|\cdot|$-grading and the $\deg$-grading on
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ coincide. Therefore:

$\nu(\phi(g)\bowtie 1)f\nu(\phi(g)\bowtie 1)^{-1}=\phi(g)\cdot f=g\cdot
f=ufu^{-1}\qquad\forall f\in{\rm End}(P).$

Since ${\rm End}(P)$ is central and $\nu$ is an algebra morphism,
$u^{\prime}:=\nu(\phi(g)\bowtie 1)=\lambda u$ with $\lambda=\pm 1$ (both
possibilities will be analyzed later). The element
$w^{\prime}:=\nu(\phi(h)\bowtie 1)$ satisfies

$\phi(h)\cdot f=w^{\prime}f-(\phi(g)\cdot f)w^{\prime}\qquad\forall f\in{\rm
End}(P).$

Thus, we can take $W^{\prime}$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that

$W^{\prime}U+UW^{\prime}=0,\quad(W^{\prime})^{2}=0\quad\phi(h)\cdot
Z=W^{\prime}Z-(g\cdot Z)W^{\prime}$

for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$ Arguing as for $W$
before, we see that $W^{\prime}=\gamma X\\#1+\delta Y\\#1$ for some
$\gamma,\delta\in k$. It follows from the last relation of $D(H_{4})$ in §1
that

$\nu(\varepsilon\bowtie hg)\nu(\phi(h)\bowtie 1)+\nu(\phi(h)\bowtie
1)\nu(\varepsilon\bowtie hg)=\nu(\phi(g)\bowtie 1)\nu(\varepsilon\bowtie
g)-\nu(\varepsilon\bowtie g)^{2}.$

This implies $WW^{\prime}+W^{\prime}W=\lambda-1.$ Besides,

$0=\phi(h)\cdot W^{\prime}=2(W^{\prime})^{2}=s\gamma+\delta l.$

Now, by direct computation:

$\begin{array}[]{rl}\lambda-1&=WW^{\prime}+W^{\prime}W\vspace{2pt}\\\
&=\alpha((X-tY)(\gamma X+\delta Y)+(\gamma X+\delta Y)(X-tY))\vspace{2pt}\\\
&=\alpha\gamma(2a-ts)+\alpha\delta(s-2t(l-b))\\\
&=-t\gamma-\delta.\end{array}$

Let us first assume $\lambda=1$. Then, $\gamma(s-tl)=0$. If $\gamma=0,$ then
$\delta=0$ and so $W^{\prime}=0$. This means that the $\phi(h)$-action is
identically zero, yielding $s=l=0$. Otherwise, $s=tl$ and we are done.

We finally show that the possibility $\lambda=-1$ can not occur. If
$\lambda=-1$, then $\delta=2-t\gamma$ and $s\gamma=-(2-t\gamma)l$. On the
other hand,

$l=\phi(h)\cdot(Y\\#1)=W^{\prime}(Y\\#1)+(Y\\#1)W^{\prime}=s\gamma+2(2-t\gamma)(l-b)$
(3.2)

Moreover,

$\begin{array}[]{rl}0&=(W^{\prime})^{2}\vspace{2pt}\\\
&=\gamma^{2}a+\delta^{2}(l-b)+\gamma\delta s\vspace{2pt}\\\
&\stackrel{{\scriptstyle(\ref{a=})}}{{=}}\gamma^{2}(ts-t^{2}(l-b))+(2-t\gamma)^{2}(l-b)+\gamma(2-t\gamma)s\vspace{2pt}\\\
&=2(l-b)(2-2t\gamma)+2\gamma s\end{array}$

From here, $s\gamma=(2t\gamma-2)(l-b).$ Substituting this in (3.2) we get
$l=2b$, contradicting the fact that $C(b;1,l)$ is $(H_{4},R_{l})$-Azumaya.

(2) If $l\not=0$, then $Im(\iota_{l})=Im(i_{l^{-1}})$ by Proposition 3.1 and
the statement follows from (1). It remains to show that $[C(a;t,s)]\in
Im(\iota_{0})$ implies $t=0$. If $[C(a;t,s)]\in Im(\iota_{0}),$ there exists
$b\in k^{\cdot}$ and an $H_{4}$-Azumaya algebra $A$ with trivial $h$-action
and trivial $\phi(h)$-action such that $[C(a;t,s)]=[A\\#C(b;0,1)].$ Then
$C(a;t,s)\\#C(-b;0,1)\\#\overline{A}\cong{\rm End}(P)$ for some
$D(H_{4})$-module $P$. Arguing as in (1) we see that there is $W=\alpha
X\\#1+\beta Y\\#1\in(C(a;t,s)\\#C(-b;0,1))\\#\overline{A}$ for some
$\alpha,\beta\in k$ such that

$\begin{array}[]{ll}&\hskip 11.0pth\cdot Z=W(g\cdot Z)-ZW,\\\ 0&=h\cdot
W=-2W^{2}=\alpha t+\beta,\\\ t&=h\cdot(X\\#1)=-2a\alpha,\\\
0&=h\cdot(Y\\#1)=2b\beta.\end{array}$

From here if follows that $t=0$. $\Box$

###### Corollary 3.4

Let $[C(a;t,s)]$, $[C(b;p,q)]$ be in $BQ(k,H_{4})$. Then
$[C(a;t,s)]=[C(b;p,q)]$ if and only if $C(a;t,s)\cong C(b;p,q)$.

Proof: We analyze the case $t\neq 0$, the other cases are treated similarly.
If $[C(a;t,s)]=[C(b;p,q)]$ and $p=0$ then $[C(a;t,s)]\in Im(\iota_{0}),$
contradicting Theorem 3.3. Then $tp\neq 0$ and we may reduce to the case
$[C(a;1,s)]=[C(b;1,q)]\in Im(i_{q})$. Applying again Theorem 3.3 we see that
$s=q$ and the equality of classes is an equality in $BM(k,H_{4},R_{q})$.
Applying $\Phi_{0}^{-1}\Psi_{q}^{-1}\Phi_{q}$ we obtain the equality
$[C(a-2^{-1}q;1,0)]=[C(b-2^{-1}q;1,0)]$ in $BM(k,H_{4},R_{0})$. From
Proposition 2.3, we obtain $(4a-2q)^{-1}=(4b-2q)^{-1}$ and we have the
statement. $\Box$

###### Theorem 3.5

Let $i_{t}:BM(k,H_{4},R_{t})\to BQ(k,H_{4})$ and
$\iota_{s}:BC(k,H_{4},r_{s})\to BQ(k,H_{4})$ be the natural embeddings in
$BQ(k,H_{4})$. Then:

  1. (1)

$Im(i_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $ts=1$. If this
is the case, then $Im(i_{t})=Im(\iota_{s})$;

  2. (2)

$Im(i_{t})\cap Im(i_{s})\neq i_{0}(BW(k))$ if and only if $t=s$;

  3. (3)

$Im(\iota_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $t=s$.

Proof: This is a consequence of Propositions 2.3, 2.5, 2.6, 3.1 and Theorem
3.3. $\Box$

##  4 The action of $Aut(H_{4})$ on $Im(i_{t})$ and $Im(\iota_{s})$

For a Hopf algebra $H$, a group morphism from ${\rm Aut}_{\rm Hopf}(H)$ to
$BQ(k,H_{4})$ has been constructed in [8], where the case of $H_{4}$ was also
analized. The image of an automorphism $\alpha$ can be represented as follows.

Let us denote by $H_{\alpha}$ the right $H$-comodule $H$ with left $H$-action
$l\cdot m=\alpha(l_{(2)})mS^{-1}(l_{(1)})$. Then $A_{\alpha}={\rm
End}(H_{\alpha})$ can be endowed of the $H$-Azumaya algebra structure:

$\begin{array}[]{l}(l\cdot f)(m)=l_{(1)}\cdot f(S(l_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=\sum f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)}.\end{array}$

The assignment $\alpha\mapsto[A_{\alpha^{-1}}]$ defines a group morphism ${\rm
Aut}_{\rm Hopf}(H)\to BQ(k,H)$. The image of ${\rm Aut}_{\rm Hopf}(H)$ acts on
$BQ(k,H)$ by conjugation. An easy description of
$[B(\alpha)]:=[A_{\alpha}][B][A_{\alpha}]^{-1}$ for any representative $B$ has
been given in [8, Theorem 4.11]. As an algebra $B(\alpha)$ coincides with $B$,
while the $H$-action and $H$-coaction are:

$h\cdot_{\alpha}b=\alpha(h)\cdot
b,\quad\rho_{\alpha}(b)=b_{(0)}\otimes\alpha^{-1}(b_{(1)}).$ (4.1)

When $H=H_{4}$ the Hopf automorphism group is ${\rm Aut}_{\rm
Hopf}(H_{4})\cong k^{\cdot}$ and consists of the morphisms that are the
identity on $g$ and multiply $h$ by a nonzero scalar $\alpha$. The module
$H_{\alpha}$ has action

$\begin{array}[]{l}g\cdot g=g,\quad g\cdot h=-h,\vspace{2pt}\\\ h\cdot
g=\alpha hg+g^{2}S^{-1}(h)=-(1+\alpha)gh,\quad h\cdot h=0,\end{array}$

and the kernel of the group morphism consists of $\\{\pm 1\\}$. We may thus
embed $(k^{\cdot})^{2}\cong k^{\cdot}/\\{\pm 1\\}$ into $BQ(k,H_{4})$ (cf.
[19]). We shall denote by $K$ the image of this group morphism.

We analyze this action on the classes and subgroups described in the previous
sections.

###### Lemma 4.1

Let $\alpha\in k^{\cdot}$. Then:

  1. (1)

$[A_{\alpha}][C(a;t,s)][A_{\alpha}]^{-1}=[C(a;\alpha t,s\alpha^{-1})]$.

  2. (2)

$K$ acts trivially on $i_{0}(BW(k))$.

In particular, $BM(k,H_{4},R_{l\alpha^{2}})$ is conjugate to
$BM(k,H_{4},R_{l})$ in $BQ(k,H_{4})$ while $BM(k,H_{4},R_{0})$ and
$BC(k,H_{4},r_{0})$ are normalized by $K$.

Proof: (1) It follows from direct computation that

$h\cdot_{\alpha}x=\alpha t,\quad g\cdot_{\alpha}x=-x,\quad\rho(x)=x\otimes
g+s\alpha^{-1}\otimes h.$

(2) Since: the action of an automorphism of $H_{4}$ is trivial on $g$; the
action of $h$ is trivial on a representative of a class in $BW(k)$; and the
comodule map on a representative $A$ of a class in $BW(k)$ has image in
$A\otimes k{\mathbb{Z}}_{2}$, the formulas in (4.1) do not modify the action
and coaction on $A$ therefore $[A]=[A_{\alpha}][A][A_{\alpha}]^{-1}$ for every
$[A]\in i_{0}(BW(k))$.

Since $Im(i_{l})$ is generated by $i_{0}(BW(k))$ and the classes $[C(a;1,l)]$,
we see that $Im(i_{l})$ is conjugate to $Im(i_{\alpha^{2}l})$ in
$BQ(k,H_{4})$. If $l=0$ we get the statement concerning $Im(i_{0})$. The
statement concerning $BC(k,H_{4},r_{0})$ follows because this group is
generated by $i_{0}(BW(k))$ and the classes $[C(a;0,1)]$. $\Box$

###### Remark 4.2

The observation that $Im(i_{0})$ is normalized by $K$ has already been proved
in [21, §4]. Lemma 4.1 should be seen as a generalization of that result.

It is shown in [18] that $(H_{4},R_{t})$ is equivalent to $(H_{4},R_{s})$ if
and only if $t=\alpha^{2}s$ for some $\alpha\in k^{\cdot}$. The above lemma
shows that the Brauer groups of type $BM$ are conjugate in $BQ(k,H_{4})$ if
the corresponding triangular structures are equivalent. This is a general
fact:

###### Proposition 4.3

Let $R$ and $R^{\prime}$ be two equivalent quasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$(\alpha\otimes\alpha)(R^{\prime})=R$. Then the images of $BM(k,H,R)$ and
$BM(k,H,R^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.

Proof: If $B$ represents an element in $BM(k,H,R)$ then there will be an
action $\cdot$ on $B$ such that the coaction $\rho$ is given by
$\rho(b)=(R^{(2)}\cdot b)\otimes R^{(1)}$ for all $b\in B$. The image of
$\alpha$ in $BQ(k,H)$ is represented by $A_{\alpha^{-1}}$. A representative of
$[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is given by the algebra $B$ with action
$h\cdot_{\alpha^{-1}}b=\alpha^{-1}(h)\cdot b$. The coaction is given by

$\rho_{\alpha}(b)=(R^{(2)}\cdot
b)\otimes\alpha(R^{(1)})=(\alpha(R^{(2)})\cdot_{\alpha}b)\otimes\alpha(R^{(1)})=R^{\prime(2)}\cdot_{\alpha}b\otimes
R^{\prime(1)},$

so the coaction on $[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is induced by
$R^{\prime}$ and $\cdot_{\alpha}$. $\Box$

For the dual statement, the proof is left to the reader.

###### Proposition 4.4

Let $r$ and $r^{\prime}$ be two equivalent coquasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$r^{\prime}(\alpha\otimes\alpha)=r$. Then the images of $BC(k,H,r)$ and
$BM(k,H,r^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.

##  5 The subgroup $BQ_{grad}(k,H_{4})$

In this section we shall analyze the classes that can be represented by
$H_{4}$-Azumaya algebras for which the gradings coming from the $g$-action and
the comodule structure coincide. They form a subgroup that will be related to
the Brauer group $BM(k,E(2),R_{N})$ of Nichols $8$-dimensional Hopf algebra
$E(2)$ with respect to the quasitriangular structure $R_{N}$ attached to the
$2\times 2$-matrix $N$ with $1$ in the $(1,2)$-entry and zero elsewhere.

Let $BQ_{grad}(k,H_{4})$ be the set of classes that can be represented by a
$H_{4}$-Azumaya algebra $A$ for which the $|\cdot|$-grading and the
$\deg$-grading coincide. In other words, the classes in $BQ_{grad}(k,H_{4})$
can be represented by $D(H_{4})$-module algebras on which the actions of $g$
and $\phi(g)$ coincide. The last defining relation of $D(H_{4})$ in Section 1
implies that the action of $h$ and $\phi(h)$ on such representatives commute.
Clearly, $BQ_{grad}(k,H_{4})$ is a subgroup of $BQ(k,H_{4})$.

###### Proposition 5.1

$BQ_{grad}(k,H_{4})$ is normalized by $K$.

Proof: Let $[A]\in BQ_{grad}(k,H_{4})$ with $|a|=\deg(a)$ for every $a\in A$
and let $[A_{\alpha}]\in K$. Then $[A_{\alpha}\\#A\\#\overline{A_{\alpha}}]$
is represented by $A$ with action and coaction determined by (4.1). Since $g$
is fixed by all Hopf automorphisms of $H_{4}$ we have

$g\cdot_{\alpha}a=g\cdot a,\quad({\rm id}\otimes\pi)\rho_{\alpha}(a)=({\rm
id}\otimes\pi)\rho(a),$

so the two gradings are not modified by conjugation by $[A_{\alpha}]$. $\Box$

The subgroup $BQ_{grad}(k,H_{4})$ consists of those classes that can be
represented by module algebras for the quotient of $D(H_{4})$ by the Hopf
ideal $I$ generated by $\phi(g)\bowtie 1-\varepsilon\bowtie g$. Let us denote
by $\pi_{I}$ the canonical projection onto $D(H_{4})/I$.

Let $E(2)$ be the Hopf algebra with generators $c,\,x_{1},\,x_{2},$ with
relations

$c^{2}=1,\quad x_{i}^{2}=0,\quad cx_{i}+x_{i}c=0,\ i=1,2,\quad
x_{1}x_{2}+x_{2}x_{1}=0,$

coproduct

$\Delta(c)=c\otimes c,\quad\Delta(x_{i})=1\otimes x_{i}+x_{i}\otimes c,$

and antipode

$S(c)=c,\quad S(x_{i})=cx_{i}.$

The Hopf algebra morphism

$\begin{array}[]{rl}T\colon D(H_{4})&\longrightarrow E(2)\\\ \phi(g)\bowtie
1&\mapsto c\\\ \varepsilon\bowtie g&\mapsto c\\\ \varepsilon\bowtie h&\mapsto
x_{1}\\\ \phi(h)\bowtie 1&\mapsto cx_{2}\end{array}$

determines a Hopf algebra isomorphism $D(H_{4})/I\cong E(2)$. The canonical
quasitriangular structure ${\cal R}$ on $D(H_{4})$ is

$\begin{array}[]{rl}{\cal R}&=\frac{1}{2}[\varepsilon\bowtie(1\otimes
1^{*}+g\otimes g^{*}+h\otimes h^{*}+gh\otimes(gh)^{*})\bowtie
1]\vspace{3pt}\\\ &\hskip
3.0pt+\frac{1}{2}[\varepsilon\bowtie(1\otimes\varepsilon+g\otimes\varepsilon+1\otimes\phi(g)-g\otimes\phi(g)\vspace{3pt}\\\
&\hskip
20.0pt+h\otimes\phi(h)+h\otimes\phi(gh)+gh\otimes\phi(h)-gh\otimes\phi(gh))\bowtie
1]\end{array}$

so $(\pi_{I}\otimes\pi_{I})({\cal R})$ is a quasitriangular structure for
$D(H_{4})/I\cong E(2)$. Applying $T\otimes T$ to ${\cal R}$ we have:

$\begin{array}[]{rl}(T\otimes T)({\cal R})&=\frac{1}{2}(1\otimes 1+1\otimes
c+c\otimes 1-c\otimes c\\\ &\hskip 20.0pt+x_{1}\otimes cx_{2}+x_{1}\otimes
x_{2}+cx_{1}\otimes cx_{2}-cx_{1}\otimes x_{2})\end{array}$ (5.1)

The quasitriangular structures on $E(n)$ were computed in [17]. They are in
bijection with $n\times n$-matrices with entries in $k$. For a given matrix
$M$ the corresponding quasitriangular structure is denoted by $R_{M}$. The map
$T$ induces a quasitriangular morphism from $(D(H_{4}),{\cal R})$ onto
$(E(2),R_{N}),$ where $N$ is the $2\times 2$-matrix with $1$ in the
$(1,2)$-entry and zero elsewhere. If $A$ is a representative of a class in
$BQ_{grad}(k,H_{4})$ on which the ideal $I$ acts trivially, then $A$ is an
$E(2)$-module algebra and the maps $F$ and $G$ on $A\otimes A$ are the same as
those induced by $R_{N}$, so $A$ is $(E(2),R_{N})$-Azumaya.

###### Theorem 5.2

The group $BM(k,E(2),R_{N})$ fits into the following exact sequence

$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$

Proof: Restriction of scalars through $T$ provides a group morphism $T^{*}$
from $BM(k,E(2),R_{N})$ to $BQ(k,H)$ whose image is $BQ_{grad}(k,H_{4})$. The
kernel of $T^{*}$ consists of those classes $[A]$ such that $A\cong{\rm
End}(P)$ as $D(H_{4})$-module algebras, for some $D(H_{4})$-module $P$. The
class $[A]$ may be non-trivial only if $g$ and $\phi(g)$ act differently on
$P$ even though they act equally on ${\rm End}(P)$. The $\phi(g)$\- and
$g$-action on ${\rm End}(P)$ are strongly inner, hence there are elements $U$
and $u$ in ${\rm End}(P)$ such that $\phi(g)\cdot f=UfU^{-1}=ufu^{-1}=g\cdot
f$ for every $f\in{\rm End}(P).$ Since ${\rm End}(P)$ is a central algebra,
$U^{2}=u^{2}=1$, $uU=Uu$. From here, $U=\pm u,$ and if $[{\rm End}(P)]\neq 1$
in $BM(k,E(2),R_{N})$ we necessarily have $U=-u$. The actions of $g$ and
$\phi(g)$ on $P$ are given by the element $u$ and $U$ respectively, so for
every non-trivial $[A]$ in ${\rm Ker}(T^{*})$ we have $A\cong{\rm End}(P)$ for
some $D(H_{4})$-module $P$ for which $g$ acts as $-\phi(g)$. We claim that
there is at most one non-trivial element in ${\rm Ker}(T^{*})$.

Given any pair of such elements ${\rm End}(P)$ and ${\rm End}(Q)$ representing
classes in ${\rm Ker}(T^{*})$ we have ${\rm End}(P)\\#{\rm End}(Q)\cong{\rm
End}(P\otimes Q)$ as $D(H_{4})$-module algebras by [7, Proposition 4.3], where
$P\otimes Q$ is a $D(H_{4})$-module. Then, the actions of $g$ and $\phi(g)$ on
$P\otimes Q$ coincide, so $P\otimes Q$ is an $E(2)$-module. Thus, $[{\rm
End}(P)][{\rm End}(Q)]$ is trivial in $BM(k,E(2),R_{N})$ for every choice of
$P$ and $Q$. Therefore, ${\rm Ker}(T^{*})$ is either trivial or isomorphic to
${\mathbb{Z}}_{2}$. The proof is completed once we provide a non-trivial
element. Let us consider $P=k^{2}$ on which $g,\,h,\,\phi(g)$ and $\phi(h)$
act via the following matrices $u,\,w,\,U,\,W$, respectively:

$u=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
w=\left(\begin{array}[]{cc}0&0\\\ -2&0\end{array}\right),\quad U=-u,\,\quad
W=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right).$

Then $P$ is a $D(H_{4})$-module but not an $E(2)$-module. On the other hand,
the $D(H_{4})$-module algebra structure on ${\rm End}(P)$ is in fact an
$E(2)$-module algebra structure:

$g\cdot f=ufu^{-1}=UfU^{-1}=\phi(g)\cdot f;$ (5.2) $h\cdot
f=wfu^{-1}+fuw,\quad\phi(h)\cdot f=Wf-UfU^{-1}W.$ (5.3)

Moreover, ${\rm End}(P)$ is $(E(2),R_{N})$-Azumaya because it is
$H_{4}$-Azumaya. We claim that the class of ${\rm End}(P)$ is not trivial in
$BM(k,E(2),R_{N})$. Indeed, if it were trivial, then the $E(2)$-action on
${\rm End}(P)$ given by $c.f=g.f$, $x_{1}.f=h.f$ and $(cx_{2}).f=\phi(h).f$
would be strongly inner. In other words, there would exist a convolution
invertible algebra morphism $p\colon E(2)\to{\rm End}(P)$ for which $l\cdot
f=\sum p(l_{(1)})fp^{-1}(l_{(2)})$ for every $l\in E(2)$. Putting
$u^{\prime}=p(c)$ we have $c.f=u^{\prime}f(u^{\prime})^{-1}=ufu^{-1}$. Since
${\rm End}(P)$ is a central simple algebra, we necessarily have
$u^{\prime}=\lambda u$ and since $(u^{\prime})^{2}=1$ we have $\lambda=\pm 1$.
Putting $w^{\prime}=p(x_{1})$ we have
$x_{1}.f=w^{\prime}fu^{\prime}-fw^{\prime}u^{\prime}$ and since
$u^{\prime}w^{\prime}=-w^{\prime}u^{\prime}$, we have $\lambda
w^{\prime}fu+\lambda fuw^{\prime}=x_{1}.f=h.f=wfu+fuw$ for every $f\in{\rm
End}(P)$. Using $uw=-wu$ we see that $(\lambda w^{\prime}-w)f=f(\lambda
w^{\prime}-w)$ so $w=\lambda w^{\prime}+\mu$ for some $\mu\in k$. Using once
more skew-commutativity of $u$ with $w$ and $w^{\prime}$ we see that $\mu=0$.

Putting $W^{\prime}=p(cx_{2})$ and using that
$u^{\prime}W^{\prime}=-W^{\prime}u^{\prime}$ we see that
$W^{\prime}f-ufuW^{\prime}=(cx_{2}).f=\phi(h).f=Wf-ufuW$ for every $f\in{\rm
End}(P)$. From here, we deduce that $u(W^{\prime}-W)=\nu\in k$. Using skew-
commutativity of $u$ with $W$ and $W^{\prime}$ we conclude that $\nu=0$ so
$W^{\prime}=W$. Then $W^{\prime}w^{\prime}-w^{\prime}W^{\prime}=\lambda(Ww-
wW)\neq 0$ so that relation $(cx_{2})x_{1}-x_{1}(cx_{2})=0$ in $E(2)$ cannot
be respected. Hence, $[{\rm End}(P)]\neq 1$ in $BM(k,E(2),R_{N})$ and ${\rm
Ker}(T^{*})\cong{\mathbb{Z}}_{2}$. $\Box$

The following proposition shows that the groups $BM(k,H_{4},R_{l})$ may be
viewed inside $BM(k,E(2),R_{N})$ and it also describes the image through
$T^{*}$ of them.

###### Proposition 5.3

For every $(\lambda,\mu)\in k\times k$ there is a group homomorphism

$\Theta_{\lambda,\mu}\colon BM(k,H_{4},R_{\lambda\mu})\to BM(k,E(2),R_{N})$

satisfying:

  1. (1)

The image of $\Theta_{0,0}$ is the subgroup isomorphic to $BW(k)$ represented
by elements with trivial $x_{1}$\- and $x_{2}$-action and
$Ker(\Theta_{0,0})\cong(k,+)$.

  2. (2)

$\Theta_{\lambda,\mu}$ is injective if and only if $(\lambda,\mu)\neq(0,0)$.

  3. (3)

For $(\lambda,\mu)\neq(0,0),$ the image of $T^{*}\Theta_{\lambda,\mu}$ is
$Im(i_{\mu\lambda^{-1}})$ if $\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$
if $\mu\neq 0$.

Proof: For every $(\lambda,\mu)\in k\times k$ the map
$\theta_{\lambda,\mu}\colon E(2)\to H_{4}$ mapping $c\to g$, $x_{1}\to\lambda
h$ and $x_{2}\to\mu h$ is a Hopf algebra projection. A direct computation
shows that
$(\theta_{\lambda,\mu}\otimes\theta_{\lambda,\mu})(R_{N})=R_{\lambda\mu}$ so
the pull-back of $\theta_{\lambda,\mu}$ induces the desired homomorphism
$\Theta_{\lambda,\mu}$.

(1) Let $(\lambda,\mu)=(0,0)$. Then any element in $BM(k,H_{4},R_{0})$ can be
written as a pair of the form $([C(a;t,0)],[B])$ for $[B]\in BW(k)$. The image
through $\Theta_{0,0}$ of such an element is $[C(a)][B]\in BW(k)$ with trivial
$x_{i}$-action on $C(a)$. Clearly, $BW(k)=Im(\Theta_{0,0})$. That
$Ker(\Theta_{0,0})$ is isomorphic to $(k,+)$ follows from the isomorphism
$BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ and the fact that $(k,+)$ is
realized as classes admitting a representative that is trivial when viewed as
a $k{\mathbb{Z}}_{2}$-module algebra.

(2) Let $(\lambda,\mu)\neq(0,0)$. If $\Theta_{\lambda,\mu}([A])=1$ then $A$ is
isomorphic to an endomorphism algebra with strongly inner $E(2)$-action. In
other words, $A\cong{\rm End}(P)$ and there is a convolution invertible
algebra map $p\colon E(2)\to A$ such that $l\cdot a=\sum
p(l_{(1)})ap^{-1}(l_{(2)})$ for every $l\in E(2),a\in A$. There are elements
$u,v,w\in A$ with $u$ invertible such that $c\cdot a=g\cdot a=uau^{-1}$,
$x_{1}\cdot a=(va-av)u=\lambda h\cdot a$ and $x_{2}\cdot a=(wa-aw)u=\mu h\cdot
a$. Then

$0=\mu x_{1}\cdot a-\lambda x_{2}\cdot a=((\mu v-\lambda w)a-a(\mu v-\lambda
w))u\quad\forall a\in A,$

and since $u$ is invertible and $A$ is central we have $\mu v-\lambda w=\eta$
for some $\eta\in k$. The relation between $v$ and $w$ gives $\eta=0$ and so
$\mu v=\lambda w.$ Thus, the same elements $u,v$ and $w$ ensure that the
$H_{4}$-action on $A$ is strongly inner. Therefore $[A]=1$ in
$BM(k,H_{4},R_{\lambda\mu})$. The converse follows from (1).

(3) Let us now assume that $(\lambda,\mu)\neq(0,0).$ It is immediate to see
that if $[A]\in BW(k)\subset BM(k,H_{4},R_{\lambda\mu})$ is represented by an
algebra with trivial $h$-action, then $\Theta_{\lambda,\mu}([A])$ is
represented by an algebra with trivial $x_{1}$\- and $x_{2}$-action. Hence
$T^{*}\Theta_{\lambda,\mu}(BM(k,H_{4},R_{\lambda\mu}))\subset i_{0}(BW(k))$
and the restriction of $T^{*}\Theta_{\lambda,\mu}$ to $BW(k)$ is an
isomorphism onto $i_{0}(BW(k))$. Let us now consider the class
$[C(a;1,\lambda\mu)]\in BM(k,H_{4},R_{\lambda\mu})$. Its image through
$\Theta_{\lambda,\mu}$ is the algebra generated by $x$ with $x^{2}=a$, with
$c\cdot x=-x$, $x_{1}\cdot x=\lambda$ and $x_{2}\cdot x=\mu$. A direct
verification shows that
$T^{*}\Theta_{\lambda,\mu}([C(a;1,\lambda\mu)])=[C(a;\lambda,\mu)]$. Then the
image of $T^{*}\Theta_{\lambda,\mu}$ is $Im(i_{\mu\lambda^{-1}})$ if
$\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$ if $\mu\neq 0$. $\Box$

Theorem 5.2 shows that one should understand $BM(k,E(2),R_{N})$ in order to
compute $BQ(k,H_{4})$. In view of Proposition 5.3, $BM(k,E(2),R_{N})$ seems to
be much more complex that the groups of type BM treated in [10, 11, 20].

##  6 Appendix

This last section is devoted to the analysis of some difficulties occurring in
the study of the structure of $(E(2),R_{N})$-Azumaya algebras. We show that
the set of classes represented by ${\mathbb{Z}}_{2}$-graded central simple
algebras (with respect to the grading induced by the $c$-action) is not a
subgroup of $BM(k,E(2),R_{N})$.

Let us consider the braiding $\psi_{VW}$ determined by $R_{N}$ between two
left $E(2)$-modules $V$ and $W$. Let $v\in V$ and $w\in W$ be homogeneous
elements with respect to the ${\mathbb{Z}}_{2}$-grading induced by the
$c$-action. By direct computation it is:

$\begin{array}[]{l}\psi_{VW}(v\otimes w)=\sum R_{N}^{(2)}\cdot w\otimes
R_{N}^{(1)}\cdot v\vspace{2pt}\\\ \hskip 48.36958pt=(-1)^{|v||w|}w\otimes
v+(-1)^{|w|+1}(-1)^{(|v|+1)(|w|+1)}(x_{2}\cdot w)\otimes(x_{1}\cdot
v).\end{array}$

If we denote by $\psi_{0}$ the braiding associated with the
${\mathbb{Z}}_{2}$-grading we have

$\psi_{VW}(v\otimes w)=\psi_{0}(v\otimes w)+(-1)^{|w|+1}\psi_{0}(x_{1}\cdot
v\otimes x_{2}\cdot w).$ (6.1)

Let $F$ and $G$ be the maps in (1.4) defining an $(E(2),R_{N})$-Azumaya
algebra $A$ and let $F_{0}$ and $G_{0}$ be the maps defining an
$(E(2),R_{0})$-Azumaya algebra, that is, the maps determining when an
$E(2)$-module algebra is ${\mathbb{Z}}_{2}$-graded central simple. It is not
hard to verify by direct computation that, for homogeneous $a,\,b,\,d\in A$
with respect to the $c$-action we have:

$F(a\\#b)(d)=F_{0}(a\\#b)(d)+(-1)^{|d|+1}F_{0}(a\\#x_{1}\cdot b)(x_{2}\cdot
d)$ (6.2) $G(a\\#b)(d)=G_{0}(a\\#b)(d)+(-1)^{|a|+1}F_{0}(x_{2}\cdot
a\\#b)(x_{1}\cdot d)$ (6.3)

Notice that if either $x_{1}$ or $x_{2}$ acts trivially, then $F=F_{0}$ and
$G=G_{0}$. So in this case, $A$ is $(E(2),R_{N})$-Azumaya if and only if it is
${\mathbb{Z}}_{2}$-graded central simple (i.e. $A$ is $(E(2),R_{0})$-Azumaya).
We will say that the $x_{i}$-action on an $E(2)$-module algebra $A$ is inner
if there exists an odd element $v\in A$ such that $x_{i}\cdot a=v(c\cdot
a)-av$ for every $a\in A$.

###### Theorem 6.1

Let $A$ be an $(E(2),R_{N})$-Azumaya algebra. The following assertions are
equivalent:

  1. (1)

The $x_{1}$-action on $A$ is inner;

  2. (2)

The $x_{2}$-action on $A$ is inner;

  3. (3)

$A$ is a ${\mathbb{Z}}_{2}$-graded central simple algebra.

In addition, the $E(2)$-action on $A$ is inner if and only if $A$ is a central
simple algebra.

Proof: (1) $\Rightarrow$ (3) Let $v_{1}\in A$ be an odd element such that
$x_{1}\cdot a=v_{1}(c\cdot a)-av_{1}$ for all $a\in A$. Applying equality
(6.2) to any homogeneous $b$ and $d$ in $A$ gives:

$\begin{array}[]{ll}F(a\\#b)(d)&=F_{0}(a\\#b)(d)+F_{0}(a\\#b)((x_{2}\cdot
d)v_{1})\vspace{2pt}\\\ &\hskip 10.0pt+(-1)^{|d|}F_{0}(a\\#bv_{1})(x_{2}\cdot
d)\end{array}$ (6.4)

This equality extends to all elements $a$ and $b$ in $A$. If $A$ were not
${\mathbb{Z}}_{2}$-graded central simple, there would exist an element
$0\neq\sum_{i}a_{i}\\#b_{i}$ in $Ker(F_{0})$. Then
$(\sum_{i}a_{i}\\#b_{i})(1\\#v_{1})=\sum_{i}a_{i}\\#b_{i}v_{1}\in Ker(F_{0})$
and for every $f$ in $A$ we would have
$F_{0}(\sum_{i}a_{i}\\#b_{i})(f)=F_{0}(\sum_{i}a_{i}\\#b_{i}v_{1})(f)=0$. It
follows from (6.4) that $\sum_{i}a_{i}\\#b_{i}\in Ker(F),$ contradicting the
injectivity of $F$.

(2) $\Rightarrow$ (3) Similarly to (1) $\Rightarrow$ (3) replacing $F$ by $G$.

(3) $\Rightarrow$ (1), (2) Suppose that $A$ is a ${\mathbb{Z}}_{2}$-graded
central simple algebra. If $A$ is a central simple algebra then the
$E(2)$-action on $A$ is inner by the Skolem-Noether theorem. If $A$ is not
central simple then it is of odd type ([13, Theorem 3.4, Definition 3.5]) and
it is $(H_{4},R_{0})$-Azumaya for the subalgebra of $E(2)$, isomorphic to
$H_{4}$ generated by $c$ and $x_{i}$. By [1, Theorem 3.4] the $x_{i}$-action
is inner.

Let us finally assume that the $E(2)$-action on $A$ is inner. Then $A$ is a
${\mathbb{Z}}_{2}$-graded central simple algebra. Since $E(2)$ acts innerly on
$A$ then it acts trivially on its center $Z(A)$. Besides it is immediately
seen that $Z(A)$ is contained in the right and left $E(2)$-center, that are
trivial because $A$ is assumed to be $E(2)$-Azumaya. Hence $Z(A)$ must be
trivial and so $A$ is also a central algebra. By the structure theorems of
${\mathbb{Z}}_{2}$-graded central simple algebras ([13, Theorem IV.3.4]), $A$
is central simple. $\Box$

###### Proposition 6.2

Let $A$ and $B$ be two equivalent $(E(2),R_{N})$-Azumaya algebras. Then the
$x_{i}$-action on $A$ is inner if and only if it is so on $B$.

Proof: Let $P$ and $Q$ be finite dimensional $E(2)$-modules for which
$A\\#{\rm End}(P)\cong B\\#{\rm End}(Q)$. If the $x_{i}$-action on $A$ is
inner then it is so on $A\\#{\rm End}(P)$ by [11, Proposition 4.6], hence it
is so on $B\\#{\rm End}(Q)$, which is a ${\mathbb{Z}}_{2}$-graded central
simple algebra by Theorem 6.1. For $i=1,2$, let $W_{i},v_{i}$ be odd elements
in $B\\#{\rm End}(Q)$ and ${\rm End}(Q)$ respectively inducing the
$x_{i}$-action. We recall that $x_{j}\cdot v_{i}=0$ because the action on
${\rm End}(Q)$ is strongly inner, while $x_{j}\cdot W_{i}$ is a scalar for
every pair $i,j$ because $x_{j}\cdot W_{i}$ belongs to the graded center of
$B\\#{\rm End}(Q)$. The odd elements $T_{i}=W_{i}-1\\#v_{i}-(x_{2}\cdot
W_{i})(1\\#v_{1})\in B\\#{\rm End}(Q)$ for $i=1,2$ are such that $x_{j}\cdot
T_{i}=x_{j}\cdot W_{i}$ for every $i$ and $j$. Moreover, for every homogeneous
$f\in{\rm End}(Q)$ with respect to the $c$-action we have:

$\begin{array}[]{l}(-1)^{|f|}T_{i}(1\\#f)=W_{i}(c\cdot 1\\#c\cdot
f)-1\\#v_{i}(c\cdot f)-(x_{2}\cdot W_{i})(1\\#v_{1}(c\cdot f))\vspace{2pt}\\\
\hskip 36.98866pt=(1\\#f)W_{i}-(1\\#fv_{i})-(x_{2}\cdot
W_{i})(1\\#fv_{1})-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)[W_{i}-1\\#v_{i}-(x_{2}\cdot W_{i})(1\\#v_{1})]-(x_{2}\cdot
W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)T_{i}-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f)).\end{array}$

In other words,

$(1\\#f)T_{i}=(-1)^{|f||T_{i}|}T_{i}(1\\#f)+(x_{2}\cdot
T_{i})(x_{1}\cdot(1\\#f)),$

so by (6.1) the element $T_{i}\in C^{l}_{B\\#{\rm End}(Q)}({\rm End}(Q)),$ the
left centralizer of ${\rm End}(Q)$ in $B\\#{\rm End}(Q)$, that is, $T_{i}\in
B\\#1$ by the double centralizer theorem [1, Theorem 2.3]. Besides, for every
homogeneous $b\in B$ we have:

$\begin{array}[]{rl}T_{i}(c\cdot
b\\#1)-(b\\#1)T_{i}&=(-1)^{|b|}W_{i}(b\\#1)-(b\\#v_{i})-(x_{2}\cdot
W_{i})(b\\#v_{1})\vspace{2pt}\\\ &\hskip
10.0pt-(b\\#1)W_{i}+(b\\#v_{i})+(x_{2}\cdot W_{i})(b\\#v_{1})\vspace{2pt}\\\
&=x_{i}\cdot(b\\#1).\end{array}$

Hence the $x_{i}$-action on $B$ is inner. $\Box$

We conclude by showing that, contrarily to the cases treated in the literature
([10, 11, 20]), a Skolem-Noether-like approach is probably not appropriate for
the computation of $BM(k,E(2),R_{N})$ because the set of classes admitting a
representative with inner action is not a subgroup.

###### Theorem 6.3

The classes in $BM(k,E(2),R_{N})$ that are represented by
${\mathbb{Z}}_{2}$-graded central simple algebras do not form a subgroup.

Proof: Let $t\neq 0,1$ and $q\neq 2$ be in $k$. We consider the representative
$C(1;t,2)$ generated by $x$ with $x^{2}=1$, $c\cdot x=-x$, $x_{1}\cdot x=t$
and $x_{2}\cdot x=2$ and the representative $C(1;1,q)$ generated by $y$ with
$y^{2}=1$, $c\cdot y=-y$, $x_{1}\cdot y=1$ and $x_{2}\cdot y=q$. Both are
$(E(2),R_{N})$-Azumaya because $C(1;1,2t)$ is $(H_{4},R_{2t})$-Azumaya,
$C(1;1,q)$ is $(H_{4},R_{q})$-Azumaya and $C(1;t,2),C(1;1,q)$ are obtained
from these ones respectively by pulling back through $\theta_{\lambda,\mu}$.
They are also ${\mathbb{Z}}_{2}$-graded central simple algebras. Their product
$C(1;t,2)\\#C(1;1,q)$ is generated by the odd elements $X$ and $Y$ with
$X^{2}=1$, $Y^{2}=1$ and $XY+YX=2$. The element $X-Y$ is easily seen to lie in
the ${\mathbb{Z}}_{2}$-graded center, so $C(1;t,2)\\#C(1;1,q)$ is not a
${\mathbb{Z}}_{2}$-graded central simple algebra. If $B$ were another
representative of $[C(1;t,2)\\#C(1;1,q)]$ that is a ${\mathbb{Z}}_{2}$-graded
central simple algebra, then by Theorem 6.1, the $x_{1}$-action on it would be
inner. By Proposition 6.2, $x_{1}$ would act innerly on $C(1;t,2)\\#C(1;1,q)$.
Applying again Theorem 6.1, $C(1;t,2)\\#C(1;1,q)$ would be
${\mathbb{Z}}_{2}$-graded central simple. $\Box$

Acknowledgements

This research was partially supported by the Azioni Integrate Italia-España
AIIS05E34A Algebre, coalgebre, algebre di Hopf e loro rappresentazioni. The
second named author is also supported by projects MTM2008-03339 from MCI and
FEDER and P07-FQM-03128 from Junta de Andalucía.

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