Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-
Categories

Generalized Clifford Algebras as Algebras
in Suitable Symmetric Linear Gr-Categories

Tao CHENG †‡, Hua-Lin HUANG † and Yuping YANG †

T. Cheng, H.-L. Huang and Y. Yang

† School of Mathematics, Shandong University, Jinan 250100, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ School of Mathematical Science, Shandong Normal University, Jinan 250014,
China

Received October 22, 2015, in final form January 06, 2016; Published online
January 12, 2016

By viewing Clifford algebras as algebras in some suitable symmetric Gr-
categories, Albuquerque and Majid were able to give a new derivation of some
well known results about Clifford algebras and to generalize them. Along the
same line, Bulacu observed that Clifford algebras are weak Hopf algebras in
the aforementioned categories and obtained other interesting properties. The
aim of this paper is to study generalized Clifford algebras in a similar
manner and extend the results of Albuquerque, Majid and Bulacu to the
generalized setting. In particular, by taking full advantage of the gauge
transformations in symmetric linear Gr-categories, we derive the decomposition
theorem and provide categorical weak Hopf structures for generalized Clifford
algebras in a conceptual and simpler manner.

generalized Clifford algebra; symmetric Gr-category; twisted group algebra

15A66; 18D10; 16S35

##  1 Introduction

Clifford algebras and their generalizations have played important roles in
various branches of science and engineering. Though they have been studied in
depth for many years, some new ideas still get involved and novel properties
and applications are found. In recent years, with the pioneering work of
Albuquerque and Majid [1, 2], Clifford algebras can be seen as algebras in
some suitable symmetric Gr-categories. Many well known results about Clifford
algebras thus can be derived by some novel ideas arising from the theory of
tensor categories and this viewpoint also helps to make natural
generalization. Along the same line, Bulacu observed that Clifford algebras
are weak Hopf algebras in the aforementioned categories and obtained other
interesting properties, see [4].

The aim of this paper is to study generalized Clifford algebras, written
simply GCAs in the following, in a similar manner and extend the results of
Albuquerque, Majid [1, 2] and Bulacu [3, 4] to the generalized setting. We
provide an explicit formula of 2-cocycles to present GCAs as twisted group
algebras. This helps us to put them inside suitable symmetric Gr-categories
and apply the working philosophy of Albuquerque–Majid. In particular, by
taking full advantage of the gauge transformations in symmetric linear Gr-
categories, we derive the decomposition theorem and provide completely the
categorical weak Hopf structures for GCAs in a conceptual and simpler manner.

The paper is organized as follows. In Section 2, we recall the notion of
symmetric linear Gr-categories and some examples of algebras inside them. The
presentation of GCAs as explicit twisted group algebras is given in Section 3.
The well known decomposition theorem of GCAs are also derived conceptually and
in a very simple manner with a help of suitable gauge transformations. Section
4 is devoted to the categorical weak Hopf structures of GCAs. The complete
determination of the weak Hopf structures of GCAs in some suitable symmetric
Gr-categories is translated to a much simpler situation via gauge
transformations. This simplifies, improves and generalizes the main results of
[3, 4].

Throughout the paper, let $\mathbbm{k}$ be a field and let $\mathbbm{k}^{*}$
denote the multiplicative group $\mathbbm{k}{\setminus}\\{0\\}$. Fix a
positive integer $n\geq 2$. Always assume that there exists in $\mathbbm{k}$ a
primitive $n$-th root of unity $\omega$ and
$\operatorname{char}\mathbbm{k}\nmid n$. By
$\mathbbm{Z}_{n}=\\{\bar{0},\bar{1},\dots,\overline{n-1}\\}$ we denote the
cyclic group of order $n$ written additively.

##  2 Symmetric linear Gr-categories and algebras therein

The aim of this section is to collect some preliminary concepts and notations
about symmetric linear Gr-categories and various algebraic structures within
them.

###  2.1 Symmetric linear Gr-categories

Let $G$ be a group with unit $e$ and $\mathbbm{k}$ a field. By
$\operatorname{Vec}_{G}$ we denote the category of finite-dimensional
$\mathbbm{k}$-spaces graded by the group $G$. Adorned with the usual tensor
product of $G$-graded vector spaces and an associativity constraint given by a
normalized 3-cocycle $\Phi$ on $G$, i.e., a function $\Phi\colon G\times
G\times G\rightarrow\mathbbm{k}^{*}$ such that for all $x,y,z,t\in G$

$\displaystyle\Phi(y,z,t)\Phi(x,yz,t)\Phi(x,y,z)=\Phi(x,y,zt)\Phi(xy,z,t)\qquad\mathrm{and}\qquad\Phi(x,e,y)=1,$

then $\operatorname{Vec}_{G}$ becomes a tensor category which is called a
$\mathbbm{k}$-linear Gr-category over $G$ and is denoted by
$\operatorname{Vec}_{G}^{\Phi}$ in the rest of the paper. When $\Phi$
vanishes, i.e., $\Phi(x,y,z)\equiv 1$, we write simply
$\operatorname{Vec}_{G}$. A linear Gr-category $\operatorname{Vec}_{G}^{\Phi}$
is said to be braided, if there exists a braiding given by a quasi-bicharacter
$\mathcal{R}$ with respect to $\Phi$, that is a function $\mathcal{R}\colon
G\times G\rightarrow\mathbbm{k}^{*}$ satisfying

$\displaystyle\mathcal{R}(xy,z)=\mathcal{R}(x,z)\mathcal{R}(y,z)\frac{\Phi(z,x,y)\Phi(x,y,z)}{\Phi(x,z,y)},$
$\displaystyle\mathcal{R}(x,yz)=\mathcal{R}(x,y)\mathcal{R}(x,z)\frac{\Phi(y,x,z)}{\Phi(y,z,x)\Phi(x,y,z)}$

for all $x,y,z\in G$. Note that a necessary condition for
$\operatorname{Vec}_{G}^{\Phi}$ being braided is that $G$ is abelian. A
symmetric Gr-category is a braided Gr-category
$(\operatorname{Vec}_{G}^{\Phi},\mathcal{R})$ in which the braiding
$\mathcal{R}$ is symmetric, in other words,
$\mathcal{R}(x,y)\mathcal{R}(y,x)=1$ for any $x,y\in G$. Note that, symmetric
braidings of $\operatorname{Vec}_{G}$ are nothing other than the familiar
symmetric bicharachers of $G$.

For more details on symmetric linear Gr-categories, the reader is referred to
[6, 7].

###  2.2 Algebras and coalgebras in Gr-categories

An algebra in the linear Gr-category $\operatorname{Vec}_{G}^{\Phi}$ is an
object $A$ with a multiplication morphism $m\colon A\otimes A\to A$ and a unit
morphism $u\colon\mathbbm{k}\to A$ satisfying the associativity and the
unitary conditions of usual associative algebras but expressed in diagrams of
the category $\operatorname{Vec}_{G}^{\Phi}$. More precisely, an algebra $A$
in $\operatorname{Vec}_{G}^{\Phi}$ is a finite-dimensional $G$-graded space
$\oplus_{g\in G}A_{g}$ with a multiplication $\cdot$ such that

$\displaystyle A_{g}\cdot A_{h}\subseteq A_{gh},\qquad(a\cdot b)\cdot
c=\Phi(|a|,|b|,|c|)a\cdot(b\cdot c)$

for all homogeneous elements $a,b,c\in A$. Here and below, $|a|$ denotes the
$G$-degree of $a$. There is also a unit element $1$ in $A$ such that

$\displaystyle 1\cdot a=a=a\cdot 1$

for all $a\in A$. Further, if the Gr-category $\operatorname{Vec}_{G}^{\Phi}$
is braided with braiding $\mathcal{R}$, then we say that two homogeneous
elements $a,b\in A$ are commutative if

$\displaystyle a\cdot b=\mathcal{R}(|a|,|b|)b\cdot a.$

If any two homogeneous elements commute, then we say that $A$ is a commutative
algebra in $\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$.

Dually, a coalgebra in the linear Gr-category $\operatorname{Vec}_{G}^{\Phi}$
is an object $C$ with a comultiplication morphism $\Delta\colon C\to C\otimes
C$ and a counit morphism $\varepsilon\colon C\to\mathbbm{k}$ satisfying the
coassociativity and counitary axioms in the category
$\operatorname{Vec}_{G}^{\Phi}$.

We remark that algebras in $\operatorname{Vec}_{G}^{\Phi}$ are also called
$G$-graded quasialgebras in [1]. In addition, morphisms for algebras and
coalgebras in $\operatorname{Vec}_{G}^{\Phi}$ can be defined in an obvious
manner and thus we omit further details.

###  2.3 Twisted group algebras

Algebras in $\operatorname{Vec}_{G}^{\Phi}$ may be viewed as a natural
generalization of the familiar twisted group algebras. Let $G$ be a group and
$F$ a normalized 2-cochain on $G$, i.e., a function $F\colon G\times
G\rightarrow\mathbbm{k}^{*}$ with $F(e,x)=F(x,e)=1$. Then we can define a
twisted group algebra $\mathbbm{k}_{F}[G]$ which has the same vector space as
the group algebra $\mathbbm{k}[G]$ but a different product twisted by $F$,
namely

$\displaystyle g\star h=F(g,h)gh,\qquad\forall\,g,h\in G.$

Clearly, $\mathbbm{k}_{F}[G]$ is an algebra in the linear Gr-category
$\operatorname{Vec}_{G}^{\partial F}$, where

$\displaystyle\partial F(x,y,z)=\frac{F(x,y)F(xy,z)}{F(x,yz)F(y,z)}$

is the differential of $F$. Note that, the twisted group algebra
$\mathbbm{k}_{F}[G]$ is associative in the usual sense if and only if $F$ is a
2-cocycle, i.e., $\partial F=0$, on $G$. Furthermore, if $G$ is abelian then
$\operatorname{Vec}_{G}^{\partial F}$ is symmetric with braiding given by

$\displaystyle\mathcal{R}_{F}(g,h)=\frac{F(g,h)}{F(h,g)},\qquad\forall\,g,h\in
G,$

and $\mathbbm{k}_{F}[G]$ is commutative in the symmetric Gr-category
$\big{(}\operatorname{Vec}_{G}^{\partial F},\mathcal{R}_{F}\big{)}$.

Many interesting algebras appear as twisted group algebras. Here we recall
some examples presented in [1, 2, 15]. Let $\mathbbm{R}$ denote the field of
real numbers, $\mathbbm{Z}_{2}=\\{0,1\\}$ the cyclic group of order 2, and
$\mathbbm{Z}_{2}^{n}$ the direct product of $n$ copies of $\mathbbm{Z}_{2}$.
Elements of $\mathbbm{Z}_{2}^{n}$ are written as $n$-tuples of $\\{0,1\\}$ and
the group product is written as $+$. Define functions
$f_{m}\colon\mathbbm{Z}_{2}^{n}\times\mathbbm{Z}_{2}^{n}\to\mathbbm{Z}_{2}$
for all $1\leq m\leq 3$ by

$\displaystyle f_{1}(x,y)=\sum_{i}x_{i}y_{i},\qquad
f_{2}(x,y)=\sum_{i<j}x_{i}y_{j},\qquad
f_{3}(x,y)=\sum\limits_{\begin{subarray}{c}{\rm distinct}\ i,\,j,\,k\\\
i<j\end{subarray}}x_{i}x_{j}y_{k}.$

  1. 1.

Let $F_{\rm
Cl}\colon\mathbbm{Z}_{2}^{n}\times\mathbbm{Z}_{2}^{n}\to\mathbbm{R}^{*}$ be a
function defined by

$\displaystyle F_{\rm Cl}(x,y)=(-1)^{f_{1}(x,y)+f_{2}(x,y)}.$

Then the associated twisted group algebra $\mathbbm{R}_{F_{\rm
Cl}}[\mathbbm{Z}_{2}^{n}]$ is the well-known real Clifford algebra ${\rm
Cl}_{0,n}$, see [2] for detail. This recovers the algebra of complex numbers
$\mathbb{C}$ when $n=1$ and the algebra of quaternions $\mathbb{H}$ when
$n=2$. Note that ${\rm Cl}_{0,n}$ is associative in the usual sense since the
function $F_{\rm Cl}$ is a 2-cocycle.

  2. 2.

Assume $n\geq 3$. Define the function
$F_{\mathbb{O}}\colon\mathbbm{Z}_{2}^{n}\times\mathbbm{Z}_{2}^{n}\to\mathbbm{R}^{*}$
by

$\displaystyle F_{\mathbb{O}}(x,y)=(-1)^{f_{1}(x,y)+f_{2}(x,y)+f_{3}(x,y)}.$

Then the twisted group algebra
$\mathbbm{R}_{F_{\mathbb{O}}}[\mathbbm{Z}_{2}^{n}]$ is the algebra of higher
octonions $\mathbb{O}_{n}$ introduced in [15] by generalizing the realization
of octonions via twisted group algebras (i.e., the case of $n=3$) observed in
[1]. Clearly, the series of algebras $\mathbb{O}_{n}$ are nonassociative in
the usual sense as the function $F_{\mathbb{O}}$ is not a 2-cocycle.

###  2.4 Braided tensor products

If $A_{1}$ and $A_{2}$ are two algebras in a symmetric linear Gr-category
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$, then there is a
natural braided tensor product algebra $A_{1}\widehat{\otimes}A_{2}$ living in
same category with product given by

$\displaystyle\big{(}a\widehat{\otimes}a^{\prime}\big{)}\big{(}b\widehat{\otimes}b^{\prime}\big{)}=\frac{\Phi(|a|,|a^{\prime}|,|b|)\Phi(|a||b|,|a^{\prime}|,|b^{\prime}|)}{\Phi(|a||a^{\prime}|,|b|,|b^{\prime}|)\Phi(|a|,|b|,|a^{\prime}|)}\mathcal{R}(|a^{\prime}|,|b|)a\cdot
b\widehat{\otimes}a^{\prime}\cdot b^{\prime},$

for all homogeneous elements $a,b\in A_{1}$ and $a^{\prime},b^{\prime}\in
A_{2}$. Moreover, if $A_{1}$, $A_{2}$, $A_{3}$ are algebras in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$, then
$\big{(}A_{1}\widehat{\otimes}A_{2}\big{)}\widehat{\otimes}A_{3}$ is naturally
isomorphic to
$A_{1}\widehat{\otimes}\big{(}A_{2}\widehat{\otimes}A_{3}\big{)}$. Thus in the
following, for brevity we will omit the parentheses for multiple braided
tensor product of algebras in symmetric tensor categories.

Dually, if $C_{1}$ and $C_{2}$ are two coalgebras in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$, then there is a
braided tensor product coalgebra $C_{1}\widehat{\otimes}C_{2}$ in the same
category with coproduct defined by

$\displaystyle\Delta\big{(}a\widehat{\otimes}b\big{)}=\frac{\Phi(|a_{1}|,|a_{2}|,|b_{1}|)\Phi(|a_{1}||b_{1}|,|a_{2}|,|b_{2}|)}{\Phi(|a_{1}||a_{2}|,|b_{1}|,|b_{2}|)\Phi(|a_{1}|,|b_{1}|,|a_{2}|)}\mathcal{R}(|a_{2}|,|b_{1}|)(a_{1}\widehat{\otimes}b_{1})\otimes(a_{2}\widehat{\otimes}b_{2}),$

for all homogeneous $a\in C_{1}$, $b\in C_{2}$. Here and in the following we
use Sweedler’s sigma notation $\Delta(a)=a_{1}\otimes a_{2}$ for coproduct.

###  2.5 Weak Hopf algebras in symmetric Gr-categories

With braided tensor product, we can define Hopf algebras and their various
generalizations in symmetric Gr-categories. Of course, one may even define
these algebraic structures in general braided tensor categories, see [12, 13].
For simplicity and for our purpose, we only recall the notion of weak Hopf
algebras in symmetric Gr-categories
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$.

######  Definition 2.1.

Call $(H,m,\mu,\Delta,\varepsilon,S)$ a weak Hopf algebra in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$, if

  1. (1)

$(H,m,\mu)$ is an algebra in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$;

  2. (2)

$(H,\Delta,\varepsilon)$ is a coalgebra in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$;

  3. (3)

$\Delta\colon H\longrightarrow H\widehat{\otimes}H$ is multiplicative, i.e.,
$\Delta(xy)=\Delta(x)\Delta(y)$, $\forall\,x,y\in H$;

  4. (4)

the following identities hold:

$\displaystyle\varepsilon((fg)h)=\Phi(|fg_{1}|,|g_{2}|,|h|)\Phi^{-1}(|f|,|g_{1}|,|g_{2}|)\varepsilon(fg_{1})\varepsilon(g_{2}h)$
$\displaystyle\hphantom{\varepsilon((fg)h)}{}=\Phi(|fg_{2}|,|g_{1}|,|h|)\Phi^{-1}(|f|,|g_{2}|,|g_{1}|)\mathcal{R}(|g_{1}|,|g_{2}|)\varepsilon(fg_{2})\varepsilon(g_{1}h)$
$\displaystyle\hphantom{\varepsilon((fg)h)=}{}\ \forall\ \mathrm{homogeneous\
elements}\ f,g,h\in H,$ (2.1) $\displaystyle(1_{11}\otimes 1_{12})\otimes
1_{2}=\Phi(|1_{1}|,|1_{2}|,|1_{(1)}|)(1_{1}\otimes 1_{2}1_{(1)})\otimes
1_{(2)}$ $\displaystyle\hphantom{(1_{11}\otimes 1_{12})\otimes
1_{2}}{}=\Phi(|1_{1}|,|1_{2}|,|1_{(1)}|)\mathcal{R}(|1_{2}|,|1_{(1)}|)(1_{1}\otimes
1_{(1)}1_{2})\otimes 1_{(2)},$ (2.2)

where we write $\Delta(1)=1_{1}\otimes 1_{2}=1_{(1)}\otimes 1_{(2)}$;

  5. (5)

$S\colon H\longrightarrow H$ is a morphism satisfying

$\displaystyle
h_{1}S(h_{2})=\Phi(|1_{1}|,|1_{2}|,|h|)\Phi^{-1}(|1_{1}|,|h|,|1_{2}|)\mathcal{R}(|1_{2}|,|h|)\varepsilon(1_{1}h)1_{2},$
$\displaystyle
S(h_{1})h_{2}=\Phi^{-1}(|h|,|1_{1}|,|1_{2}|)\Phi(|1_{1}|,|h|,|1_{2}|)\mathcal{R}(|h|,|1_{1}|)1_{1}\varepsilon(h1_{2}),$
$\displaystyle S(h)=(S(h_{1})h_{2})S(h_{3})$

for all homogeneous elements $h\in H$.

Note that a weak Hopf algebra in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$ is a Hopf algebra in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$ if and only if the
comultiplication is unit-preserving, and if and only if the counit is a
morphism of algebras in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$. Moreover, if
$H_{1}$ and $H_{2}$ are (weak) Hopf algebras in
$\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}$, then so is
$H_{1}\widehat{\otimes}H_{2}$.

###  2.6 Gauge transformations

When considering algebras in symmetric linear Gr-categories, we are entitled
to apply the powerful idea of gauge transformations invented for general
quasi-Hopf algebras by Drinfeld [5]. In fact, this point of view is crucial in
the pioneering works of Albuquerque and Majid [1, 2]. For the purpose of the
present paper, it suffices to recall gauge transformations in the following
special situation.

Let $G$ be a finite abelian group and $\Phi$ a 3-cocycle on $G$. It is well
known that $\mathbbm{k}[G]$ has a natural Hopf algebra structure. If we extend
$\Phi$ trilinearly to $(\mathbbm{k}[G])^{\otimes 3}$, then we have a dual
quasi-Hopf algebra $(\mathbbm{k}[G],\Phi)$ where $\Phi$ is an associator, see
[1]. In this terminology, the linear Gr-category
$\operatorname{Vec}_{G}^{\Phi}$ is exactly the comodule category of
$(\mathbbm{k}[G],\Phi)$. By definition, a gauge transformation, or a twisting,
on the dual quasi-Hopf algebra $(\mathbbm{k}G,\Phi)$ is a convolution-
invertible linear map
$F\colon\mathbbm{k}[G]\otimes\mathbbm{k}[G]\to\mathbbm{k}$ satisfying
$F(g,e)=F(e,g)=1$ for all $g\in G$. In other words, $F$ is a bilinear
expansion of a normalized 2-cochain on $G$. Given a gauge transformation $F$
on $Q=(\mathbbm{k}[G],\Phi)$, one can define a new dual quasi-Hopf algebra
$Q_{F}=(\mathbbm{k}[G],\Phi\partial F)$, see [1] for more details. The
corresponding linear Gr-category of $(\mathbbm{k}[G],\Phi_{F})$ is
$\operatorname{Vec}_{G}^{\Phi\partial F}$, which is tensor equivalent to
$\operatorname{Vec}_{G}^{\Phi}$. The tensor equivalence is given by
$(\mathcal{F},\varphi_{0},\varphi_{2})\colon\operatorname{Vec}_{G}^{\Phi}\longrightarrow\operatorname{Vec}_{G}^{\Phi\partial
F}$ with

$\displaystyle\mathcal{F}(U)=U,\qquad\varphi_{0}=\operatorname{Id}_{\mathbbm{k}},\qquad\varphi_{2}\colon\
U\otimes V\longrightarrow U\otimes V,\qquad u\otimes v\mapsto F(x,y)u\otimes
v$

for all $U,V$ and $u\in U_{x}$, $v\in V_{y}$. The tensor equivalence
$\mathcal{F}$ also induces a one-to-one correspondence between the set of
braidings of $\operatorname{Vec}_{G}^{\Phi}$ and that of
$\operatorname{Vec}_{G}^{\Phi\partial F}$. Given a braiding $\mathcal{R}$ of
$\operatorname{Vec}_{G}^{\Phi}$, define $\mathcal{F}(\mathcal{R})$ by

$\displaystyle\mathcal{F}(\mathcal{R})(x,y)=\frac{F(x,y)}{F(y,x)}\mathcal{R}(x,y),\qquad\forall\,x,y\in
G.$

Then $\mathcal{F}(\mathcal{R})$ is a braiding of
$\operatorname{Vec}_{G}^{\Phi\partial F}$ and

$\displaystyle(\mathcal{F},\varphi_{0},\varphi_{2})\colon\
\big{(}\operatorname{Vec}_{G}^{\Phi},\mathcal{R}\big{)}\longrightarrow\big{(}\operatorname{Vec}_{G}^{\Phi\partial
F},\mathcal{F}(\mathcal{R})\big{)}$

is a braided tensor equivalence. The reader is referred to [1, 9, 13] for
unexplained concepts and notations.

Clearly, if $A$ is an algebra in $\operatorname{Vec}_{G}^{\Phi}$, then
$A_{F}:=\mathcal{F}(A)$ is an algebra in $\operatorname{Vec}_{G}^{\Phi\partial
F}$. Note that the multiplication of $A_{F}$ is given by

$\displaystyle x\cdot_{{}_{F}}y=F(|x|,|y|)x\cdot y$

for all homogeneous $x,y\in A$. Similarly, if $C$ is a coalgebra in
$\operatorname{Vec}_{G}^{\Phi}$, then $C_{F}:=\mathcal{F}(C)$ is a coalgebra
in $\operatorname{Vec}_{G}^{\Phi\partial F}$ with twisted comultiplication

$\displaystyle\Delta_{F}(u)=F(|u_{1}|,|u_{2}|)^{-1}u_{1}\otimes_{F}u_{2}$

for all homogeneous $u\in C$. Moreover, if $H$ is a (weak) Hopf algebra in
$\operatorname{Vec}_{G}^{\Phi}$, then $H_{F}:=\mathcal{F}(H)$ is a (weak) Hopf
algebra in $\operatorname{Vec}_{G}^{\Phi\partial F}$, etc. The merit of these
gauge transformations lies in that, studying an algebra $A$ in
$\operatorname{Vec}_{G}^{\Phi}$ amounts to studying the algebra $A_{F}$ in
$\operatorname{Vec}_{G}^{\Phi\partial F}$. By choosing suitable twisting $F$,
there might be a chance to gauge transform an algebra $A$ in
$\operatorname{Vec}_{G}^{\Phi}$ to a simpler algebra $A_{F}$ in
$\operatorname{Vec}_{G}^{\Phi\partial F}$.

##  3 GCAs as algebras in symmetric Gr-categories

In this section, we will show how to regard generalized Clifford algebras as
algebras in some suitable symmetric linear Gr-categories and apply this point
of view to derive, in an easy manner, the well known theorem of decomposition
for GCAs via gauge transformations.

###  3.1 Generalized Clifford algebras

Various notions of GCAs appear in the literature, see for example [8, 10, 11,
14, 16, 17, 18]. In the present paper, we adopt the following concept which
seems to be of the most general form. Assume
$q_{1},q_{2},\dots,q_{m}\in\mathbbm{k}^{*}$. By definition, the generalized
Clifford algebra $C^{(n)}(q_{1},q_{2},\dots,q_{m})$ is the associative
$\mathbbm{k}$-algebra with unit $1$ generated by $e_{1},e_{2},\dots,e_{m}$
subject to the relations

$\displaystyle e_{i}^{n}=q_{i}1,\qquad\forall\,i\qquad\mathrm{and}\qquad
e_{i}e_{j}=\omega e_{j}e_{i},\qquad\forall\,i>j.$

###  3.2 Formulas of 2-cocycles

By $\mathbbm{Z}_{n}^{m}$ we denote the group
$\underbrace{\mathbbm{Z}_{n}\times\cdots\times\mathbbm{Z}_{n}}_{m}$ and by
$[a]$ the integer part of a rational number $a$. Let $0\leq x^{\prime}\leq
n-1$ denote the residue of the integer $x$ divided by $n$. Elements
$g\in\mathbbm{Z}_{n}^{m}$ are written in the form $(g_{1},g_{2},\dots,g_{m})$
and the product of $\mathbbm{Z}_{n}^{m}$ is written as $+$.

Take a primitive $n$-th root of unity $\omega$ and fix
$q_{1},\dots,q_{m}\in\mathbbm{k}^{*}$. Define
$F\colon\mathbbm{Z}_{n}^{m}\times\mathbbm{Z}_{n}^{m}\rightarrow\mathbbm{k}^{*}$
by

$\displaystyle F_{\rm GCA}(g,h)=\omega^{\sum\limits_{1\leq j<i\leq
m}g_{i}h_{j}}\prod_{i=1}^{m}q_{i}^{[\frac{g_{i}+h_{i}}{n}]}$ (3.1)

for all $g,h\in\mathbbm{Z}_{n}^{m}$. It is easy to see that $F(g,0)=F(0,g)=1$,
and that

$\displaystyle\partial F_{\rm GCA}(f,g,h)=\frac{F_{\rm GCA}(f,g)F_{\rm
GCA}(f+g,h)}{F_{\rm GCA}(g,h)F_{\rm GCA}(f,g+h)}$
$\displaystyle\hphantom{\partial F_{\rm
GCA}(f,g,h)}{}=\prod_{i=1}^{m}q_{i}^{[\frac{f_{i}+g_{i}}{n}]+[\frac{(f_{i}+g_{i})^{\prime}+h_{i}}{n}]-[\frac{g_{i}+h_{i}}{n}]-[\frac{f_{i}+(g_{i}+h_{i})^{\prime}}{n}]}=1.$

Hence $F_{\rm GCA}$ is a 2-cocycle on $\mathbbm{Z}_{n}^{m}$. Define

$\displaystyle\mathcal{R}_{\rm GCA}(g,h)=\frac{F_{\rm GCA}(g,h)}{F_{\rm
GCA}(h,g)}=\omega^{\sum\limits_{1\leq j<i\leq m}(g_{i}h_{j}-g_{j}h_{i})}$
(3.2)

and clearly it is a symmetric braiding in
$\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}}$. By Section 2.3, the twisted group
algebra $\mathbbm{k}_{F_{\rm GCA}}[\mathbbm{Z}_{n}^{m}]$ is a commutative
algebra in the symmetric Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R}_{\rm GCA})$. For the
convenience of exposition, we write $u_{g}$ for the element of
$\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ corresponding to $g\in\mathbbm{Z}_{n}^{m}$.
Then the product on $\mathbbm{k}_{F_{\rm GCA}}[\mathbbm{Z}_{n}^{m}]$ is given
by

$\displaystyle u_{g}\star u_{h}=F_{\rm GCA}(g,h)u_{g+h},$

where $g,h\in\mathbbm{Z}_{n}^{m}$ and
$g+h=((g_{1}+h_{1})^{\prime},(g_{2}+h_{2})^{\prime},\dots,(g_{m}+h_{m})^{\prime})\in\mathbbm{Z}_{n}^{m}$.

###  3.3 GCAs as twisted group algebras

Keep all the notations of the previous two subsections. Now we have an
explicit realization of $C^{(n)}(q_{1},q_{2},\dots,q_{m})$ as a twisted group
algebra.

######  Proposition 3.1.

$\mathbbm{k}_{F_{\rm GCA}}[\mathbbm{Z}_{n}^{m}]$ and
$C^{(n)}(q_{1},q_{2},\dots,q_{m})$ are isomorphic as associative
$\mathbbm{k}$-algebras. Consequently, the latter is an algebra in the
symmetric Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}^{m}_{n}},\mathcal{R}_{\rm GCA})$.

###### Proof.

It is easy to see that $C^{(n)}(q_{1},q_{2},\dots,q_{m})$ has a basis
$\big{\\{}e_{1}^{g_{1}}e_{2}^{g_{2}}\cdots
e_{m}^{g_{m}}\,|\,g_{i}\in\mathbbm{Z}_{n},\,i=1,2,\dots,m\big{\\}}$. Define a
map

$\displaystyle\psi\colon\ \mathbbm{k}_{F_{\rm GCA}}[\mathbbm{Z}_{n}^{m}]$
$\displaystyle\longrightarrow C^{(n)}(q_{1},q_{2},\dots,q_{m}),$
$\displaystyle u_{g}$ $\displaystyle\to e_{1}^{g_{1}}\cdots e_{m}^{g_{m}}.$

Clearly $\psi$ is a linear isomorphism, so it remains to verify that $\psi$ is
also multiplicative. By direct computation, we have

$\displaystyle\psi(u_{g}\star u_{h})=\psi(F_{\rm GCA}(g,h)u_{g+h})=F_{\rm
GCA}(g,h)\psi(u_{g+h})$ $\displaystyle\hphantom{\psi(u_{g}\star
u_{h})}{}=\omega^{\sum\limits_{1\leq i<j\leq
m}g_{j}h_{i}}\prod_{i=1}^{m}q_{i}^{[\frac{g_{i}+h_{i}}{n}]}e_{1}^{(g_{1}+h_{1})^{\prime}}\cdots
e_{m}^{(g_{m}+h_{m})^{\prime}}$ $\displaystyle\hphantom{\psi(u_{g}\star
u_{h})}{}=\big{(}e_{1}^{g_{1}}\cdots
e_{m}^{g_{m}}\big{)}\big{(}e_{1}^{h_{1}}\cdots
e_{m}^{h_{m}}\big{)}=\psi(u_{g})\psi(u_{h}).$

Hence we have proved that $\psi$ is an isomorphism of algebras. ∎

###  3.4 Decomposition theorem of GCAs

In this subsection, we apply Proposition 3.1 to derive the well known
decomposition theorem of generalized Clifford algebras via gauge
transformations.

######  Proposition 3.2.

We have the following isomorphisms of algebras in
$(\operatorname{Vec}_{\mathbbm{Z}^{m}_{n}},\mathcal{R}_{\rm GCA})$:

$\displaystyle C^{(n)}(q_{1},\dots,q_{m})\cong
C^{(n)}(q_{1},\dots,q_{l})\widehat{\otimes}C^{(n)}(q_{l+1},\dots,q_{m})\cong
C^{(n)}(q_{1})\widehat{\otimes}C^{(n)}(q_{2})\cdots\widehat{\otimes}C^{(n)}(q_{m}).$

###### Proof.

Recall that $F_{\rm GCA}$ is the function defined in (3.1) and

$\displaystyle\mathbbm{k}_{F_{\rm GCA}}[\mathbbm{Z}_{n}^{m}]\cong
C^{(n)}(q_{1},q_{2},\dots,q_{m})$

in $(\operatorname{Vec}_{\mathbbm{Z}^{m}_{n}},\mathcal{R}_{\rm GCA})$. Let
$F_{\rm GCA}^{-1}$ be the inverse of $F_{\rm GCA}$, namely, $F_{\rm
GCA}^{-1}(g,h)=\frac{1}{F_{\rm GCA}(g,h)}$,
$\forall\,g,h\in\mathbbm{Z}_{n}^{m}$. Then $F_{\rm GCA}^{-1}$ is a twisting on
the dual quasi-Hopf algebra $(\mathbbm{k}_{F_{\rm
GCA}}[\mathbbm{Z}_{n}^{m}],\partial F_{\rm GCA})$ and it induces a braided
tensor equivalence:

$\displaystyle\big{(}\mathcal{F}^{-1},\varphi_{0},\varphi_{2}\big{)}\colon\
(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R})\longrightarrow(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o),$

where $o(g,h)\equiv 1$. Let $\mathcal{F}$ denote the inverse functor of
$\mathcal{F}^{-1}$. By Section 2.6, considering $\mathbbm{k}_{F_{\rm
GCA}}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R})$ amounts to
considering $\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$. Note that
$\mathbbm{k}[\mathbbm{Z}_{n}^{l}]$ with $l<m$ can be viewed as an algebra in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ by a natural manner, for
example one may assign the element $(g_{1},\dots,g_{l})\in\mathbbm{Z}_{n}^{l}$
the grading $(g_{1},\dots,g_{l},0,\dots,0)\in\mathbbm{Z}_{n}^{m}$. Thus, by
assigning appropriate gradings in this way, one has immediately in
$\left(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o\right)$ the following
isomorphisms of algebras

$\displaystyle\mathbbm{k}[\mathbbm{Z}_{n}^{m}]\cong\mathbbm{k}\big{[}\mathbbm{Z}_{n}^{l}\big{]}\otimes\mathbbm{k}\big{[}\mathbbm{Z}_{n}^{m-l}\big{]}\cong\mathbbm{k}[\mathbbm{Z}_{n}]\otimes\cdots\otimes\mathbbm{k}[\mathbbm{Z}_{n}].$
(3.3)

Note that the braiding $o$ is trivial, hence the braided tensor product in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ is the usual tensor product and
so we omit the “hats” in the preceding equation. Now by applying $\mathcal{F}$
to (3.3), the claimed isomorphisms follow. ∎

##  4 Weak Hopf structures of GCAs in symmetric Gr-categories

In this section, we will show that GCAs have various weak Hopf structures if
they are putting in suitable symmetric linear Gr-categories. With a help of
the handy gauge transformations, we generalize and simplify the results
obtained by Bulacu [3, 4]. More precisely, according to Sections 2.6 and 3.4
the study of $C^{n}(q_{1},\dots,q_{m})$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R}_{\rm GCA})$ amounts to
that of $\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$. Apparently, the latter
situation seems much simpler. Therefore, we study the weak Hopf structures of
$\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ first and then translate to
those of $C^{n}(q_{1},\dots,q_{m})$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R}_{\rm GCA})$ via gauge
transformations.

###  4.1 Weak Hopf structures of $\boldsymbol{\mathbbm{k}[\mathbbm{Z}_{n}]}$
in $\boldsymbol{(\operatorname{Vec}_{\mathbbm{Z}_{n}},o)}$

For a better exposition of our idea, we start with the simplest case in which
$m=1$. In order to find categorical weak Hopf structures on
$\mathbbm{k}[\mathbbm{Z}_{n}]$, it suffices to determine its compatible
coalgebra structures in $(\operatorname{Vec}_{\mathbbm{Z}_{n}},o)$. This can
be done by direct computation in accordance with the axiom of weak Hopf
algebras as follows.

Recall that the elements of $\mathbbm{Z}_{n}$ are denoted by $\bar{i}$, i.e.,
integers modulo $n$, and the multiplication is written additively. According
to Definition 2.1(2), the comultiplication must be of the following form

$\displaystyle\Delta(\bar{i})=\sum_{\bar{j}}\delta(\bar{j},\overline{i-j})\bar{j}\otimes\overline{i-j}.$

The coassociativity of $\Delta$ implies that

$\displaystyle\delta(\bar{i},\overline{j+k})\delta(\bar{j},\bar{k})=\delta(\bar{i},\bar{j})\delta(\overline{i+j},\bar{k}),\qquad\forall\,\bar{i},\bar{j},\bar{k}.$

In other words, $\delta$ is a 2-cocycle on $\mathbbm{Z}_{n}$. In addition, the
counitary condition says

$\displaystyle\varepsilon(\bar{0})\delta(\bar{i},\bar{0})=1=\varepsilon(\bar{0})\delta(\bar{0},\bar{i}),\qquad\forall\,\bar{i}\qquad\mathrm{and}\qquad\varepsilon(\bar{j})=0,\qquad\forall\,\bar{j}\neq\bar{0}.$
(4.1)

The multiplicity of $\Delta$ impose on $\delta$ the following

$\displaystyle\delta(\bar{k},\overline{i+j-k})=\sum_{\bar{l}}\delta(\bar{l},\overline{i-l})\delta(\overline{k-l},\overline{j+l-k}),\qquad\forall\,\bar{i},\bar{j},\bar{k}.$
(4.2)

By (2.2), the 2-cocycle $\delta$ satisfies a further condition

$\displaystyle\delta(\bar{i},\overline{j-i})=\delta(\bar{i},\overline{-i}),\qquad\forall\,\bar{i},\bar{j}.$
(4.3)

Now it follows immediately from (4.1) and (4.3) that

$\displaystyle\delta(\bar{i},\bar{j})\equiv c,\qquad\forall\,\bar{i},\bar{j}$

where $c\in\mathbbm{k}^{*}$ is a constant. Then by (4.2), we have $c=nc^{2}$,
and thus $c=\frac{1}{n}$. With this, the counit is also completely determined,
namely,

$\displaystyle\varepsilon(\bar{0})=n,\qquad\varepsilon(\bar{i})=0,\qquad\forall\,\bar{i}\neq\bar{0}.$

It is clear that $\varepsilon$ satisfies (2.1). Therefore, endowed with the
preceding comultiplication $\Delta$ and counit $\varepsilon$, the algebra
$\mathbbm{k}[\mathbbm{Z}_{n}]$ becomes a weak bialgebra in the category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}},o)$. Finally, it is not hard to observe
that, if we define a $\mathbbm{k}$-linear map
$S\colon\mathbbm{k}[\mathbbm{Z}_{n}]\longrightarrow\mathbbm{k}[\mathbbm{Z}_{n}]$
by $S(\bar{i})=\bar{i}$, $\forall\,\bar{i}$, then $S$ satisfies the antipode
axiom of weak Hopf algebras (see Definition 2.1(5)) and
$\mathbbm{k}[\mathbbm{Z}_{n}]$ becomes a weak Hopf algebra in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}},o)$.

To summarize, we have

######  Proposition 4.1.

The group algebra $\mathbbm{k}[\mathbbm{Z}_{n}]$ has a unique weak Hopf
algebra structure in the Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}},o)$ with comultiplication $\Delta$,
counit $\varepsilon$ and antipode $S$ given by

$\displaystyle\Delta(\bar{i})=\frac{1}{n}\sum_{\bar{j}}\bar{j}\otimes\overline{i-j},\qquad\varepsilon(\bar{i})=n\delta_{\bar{i},\bar{0}},\qquad
S(\bar{i})=\bar{i},\qquad\forall\,\bar{i}\in\mathbbm{Z}_{n}$

where $\delta_{\bar{i},\bar{0}}$ is the Kronecker’s delta.

###  4.2 Weak Hopf structures of
$\boldsymbol{\mathbbm{k}[\mathbbm{Z}_{n}^{m}]}$ in
$\boldsymbol{(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)}$

Now we extend the above arguments to the general situation. Recall that the
elements $g\in\mathbbm{Z}_{n}^{m}$ are written in the form
$(g_{1},g_{2},\dots,g_{m})$ and the product of $\mathbbm{Z}_{n}^{m}$ is
written as $+$. By a similar argument, any comultiplication of
$\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ is determined by a 2-cocycle
$\delta\in\operatorname{Z}^{2}(\mathbbm{Z}_{n}^{m},\mathbbm{k}^{*})$, that is,

$\displaystyle\Delta(u_{g})=\sum_{h\in\mathbbm{Z}_{n}^{m}}\delta(h,g-h)u_{h}\otimes
u_{g-h}$

and the weak bialgebra axioms in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ will impose on $\delta$ the
following conditions:

$\displaystyle\delta(f,g+h)\delta(g,h)=\delta(f,g)\delta(f+g,h),\qquad\forall\,f,g,h,$
(4.4)
$\displaystyle\varepsilon(0)\delta(g,0)=1=\varepsilon(0)\delta(0,g),\qquad\forall\,g\qquad\mathrm{and}\qquad\varepsilon(h)=0,\qquad\forall\,h\neq
0,$ (4.5)
$\displaystyle\delta(h,f+g-h)=\sum_{l}\delta(l,f-l)\delta(h-l,g+l-h),\qquad\forall\,f,g,h,$
(4.6) $\displaystyle\delta(g,h-g)=\delta(g,-g),\qquad\forall\,g,h.$ (4.7)

As before, it is clear by (4.5)–(4.7) that

$\displaystyle\delta(g,h)\equiv\frac{1}{n^{m}},\qquad\forall\,g,h,\qquad\varepsilon(g)=n^{m}\delta_{g,0},\qquad\forall\,g.$

Again, let
$S\colon\mathbbm{k}[\mathbbm{Z}_{n}^{m}]\longrightarrow\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$
be the identity map. It is routine to verify that
$(\mathbbm{k}[\mathbbm{Z}_{n}^{m}],\Delta,\varepsilon,S)$ makes a weak Hopf
algebra in $(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$. This provides the
main result of this section.

######  Theorem 4.2.

The group algebra $\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ has a unique weak Hopf
algebra structure in the Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$ with comultiplication $\Delta$,
counit $\varepsilon$ and antipode $S$ given by

$\displaystyle\Delta(u_{g})=\frac{1}{n^{m}}\sum_{h}u_{h}\otimes
u_{g-h},\qquad\varepsilon(u_{g})=n^{m}\delta_{g,0},\qquad
S(g)=g,\qquad\forall\,g\in\mathbbm{Z}_{n}^{m}.$

###  4.3 Categorical weak Hopf structures of GCAs

With a help of the results in Section 3, now we can apply the handy gauge
transformation to simplify, improve and generalize the results of Bulacu [3,
4].

We start with the following

######  Corollary 4.3.

The twisted group algebra $\mathbbm{k}_{F}[\mathbbm{Z}_{n}^{m}]$ has a unique
weak Hopf algebra structure in the symmetric Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}}^{\partial F},\mathcal{R}_{F})$ with
comultiplication $\Delta$, counit $\varepsilon$ and antipode $S$ given by

$\displaystyle\Delta(u_{g})=\frac{1}{n^{m}}\sum_{h}F(h,g-h)^{-1}u_{h}\otimes
u_{g-h},\\!\qquad\varepsilon(u_{g})=n^{m}\delta_{g,0},\\!\qquad
S(g)=g,\\!\qquad\forall\,g\in\mathbbm{Z}_{n}^{m}.$

###### Proof.

By applying the gauge transformation

$\displaystyle(\mathcal{F},\varphi_{0},\varphi_{2})\colon\
(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)\longrightarrow\big{(}\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}}^{\partial
F},\mathcal{R}_{F}\big{)},$

one has
$\mathbbm{k}_{F}[\mathbbm{Z}_{n}^{m}]=\mathcal{F}(\mathbbm{k}[\mathbbm{Z}_{n}^{m}])$
and therefore the set of weak Hopf structures of
$\mathbbm{k}_{F}[\mathbbm{Z}_{n}^{m}]$ in
$\big{(}\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}}^{\partial
F},\mathcal{R}_{F}\big{)}$ is in one-to-one correspondence with that of
$\mathbbm{k}[\mathbbm{Z}_{n}^{m}]$ in
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},o)$. Now the claim follows
immediately by the above Theorem 4.2. ∎

With this, we may discover the categorical weak Hopf structures in suitable
symmetric Gr-categories of some well known algebras by properly specifying the
twisted functions. In particular, if we take $F=F_{\rm GCA}$, then we
determine the categorical weak Hopf structure of generalized Clifford algebras
via Proposition 3.1.

######  Theorem 4.4.

The generalized Clifford algebra $C^{n}(q_{1},\dots,q_{m})$ has a unique weak
Hopf algebra structure in the symmetric Gr-category
$(\operatorname{Vec}_{\mathbbm{Z}_{n}^{m}},\mathcal{R}_{F_{\rm GCA}})$, where

$\displaystyle F_{\rm GCA}(g,h)=\omega^{\sum\limits_{1\leq j<i\leq
m}g_{i}h_{j}}\prod_{i=1}^{m}q_{i}^{[\frac{g_{i}+h_{i}}{n}]},$

with comultiplication $\Delta$, counit $\varepsilon$ and antipode $S$ given by

$\displaystyle\Delta\big{(}e_{1}^{g_{1}}e_{2}^{g_{2}}\cdots
e_{m}^{g_{m}}\big{)}=\frac{1}{n^{m}}\sum_{h}\omega^{-\sum\limits_{1\leq
j<i\leq m}h_{i}(g_{j}-h_{j})^{\prime}}$
$\displaystyle\hphantom{\Delta\big{(}e_{1}^{g_{1}}e_{2}^{g_{2}}\cdots
e_{m}^{g_{m}}\big{)}=}{}\times\prod_{i=1}^{m}\mathbf{q}_{i}e_{1}^{h_{1}}e_{2}^{h_{2}}\cdots
e_{m}^{h_{m}}\otimes
e_{1}^{(g_{1}-h_{1})^{\prime}}e_{2}^{(g_{2}-h_{2})^{\prime}}\cdots
e_{m}^{(g_{m}-h_{m})^{\prime}},$
$\displaystyle\varepsilon\big{(}e_{1}^{g_{1}}\cdots
e_{m}^{g_{m}}\big{)}=n^{m}\delta_{g,0},\qquad\mathrm{and}\qquad
S\big{(}e_{1}^{g_{1}}\cdots e_{m}^{g_{m}}\big{)}=e_{1}^{g_{1}}\cdots
e_{m}^{g_{m}},$

for all $e_{1}^{g_{1}}e_{2}^{g_{2}}\cdots e_{m}^{g_{m}}\in
C^{n}(q_{1},\dots,q_{m})$ and all $g=(g_{1},g_{2},\dots,g_{m})$,
$h=(h_{1},\dots,h_{m})\in\mathbbm{Z}_{n}^{m}$, where

$\displaystyle\mathbf{q}_{i}=\begin{cases}q_{i},&h_{i}>g_{i},\\\
1,&\text{otherwise}.\end{cases}$

###### Proof.

Direct consequence of Proposition 3.1 and Corollary 4.3. ∎

###  4.4 Final remarks

In fact, our above discussion can be easily extended to the full generality.
Namely, let $G$ be an arbitrary finite abelian group written multiplicatively
with unit $e$, let $\mathbbm{k}$ be a field with
$\operatorname{char}\mathbbm{k}\nmid|G|$, and let $F$ be an arbitrary
2-cochain on $G$. Then the twisted group algebra $\mathbbm{k}_{F}[G]$ has a
unique weak Hopf structure in the symmetric Gr-category
$\big{(}\operatorname{Vec}_{G}^{\partial F},\mathcal{R}_{F}\big{)}$ with
comultiplication $\Delta$, counit $\varepsilon$ and antipode $S$ given by

$\displaystyle\Delta(x)=\frac{1}{|G|}\sum_{u}F\big{(}u,u^{-1}x\big{)}^{-1}u\otimes
u^{-1}x,\qquad\varepsilon(x)=|G|\delta_{x,e},\qquad S(x)=x,\qquad\forall\,x\in
G.$

If we specify $G=\mathbbm{Z}_{2}^{n}$ and $F$ the Albuquerque–Majid cochain
$F_{\rm CD}$ of Cayley–Dickson algebras [1], then we determine completely the
weak Hopf structures of Cayley–Dickson algebras in
$\big{(}\operatorname{Vec}_{\mathbbm{Z}_{2}^{n}}^{\partial F_{\rm
CD}},\mathcal{R}_{F_{\rm CD}}\big{)}$. This improves the main result of Bulacu
[3]. If we specify $G=\mathbbm{Z}_{2}^{n}$ and $F=F_{\mathbb{O}}$, the Morier-
Genoud–Ovsienko cochain of higher octonions [15], then we determine completely
the weak Hopf structures of higher octonions $\mathbb{O}_{n}$ in the symmetric
Gr-category $\big{(}\operatorname{Vec}_{\mathbbm{Z}_{2}^{n}}^{\partial
F_{\mathbb{O}}},\mathcal{R}_{F_{\mathbb{O}}}\big{)}$.

### Acknowledgements

This research was supported by SRFDP 20130131110001, NSFC 11471186, NSFC
11571199, and SDNSF ZR2013AM022.

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