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

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.


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 k be a field and let k * denote the multiplicative group k\{0}. Fix a positive integer n ≥ 2. Always assume that there exists in k a primitive n-th root of unity ω and char k n. By Z n = {0,1, . . . , n − 1} we denote the cyclic group of order n written additively.

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.

Symmetric linear Gr-categories
Let G be a group with unit e and k a field. By Vec G we denote the category of finite-dimensional 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 Φ on G, i.e., a function and Φ(x, e, y) = 1, then Vec G becomes a tensor category which is called a k-linear Gr-category over G and is denoted by Vec Φ G in the rest of the paper. When Φ vanishes, i.e., Φ(x, y, z) ≡ 1, we write simply Vec G . A linear Gr-category Vec Φ G is said to be braided, if there exists a braiding given by a quasi-bicharacter R with respect to Φ, that is a function R : for all x, y, z ∈ G. Note that a necessary condition for Vec Φ G being braided is that G is abelian. A symmetric Gr-category is a braided Gr-category (Vec Φ G , R) in which the braiding R is symmetric, in other words, R(x, y)R(y, x) = 1 for any x, y ∈ G. Note that, symmetric braidings of 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].

Algebras and coalgebras in Gr-categories
An algebra in the linear Gr-category Vec Φ G is an object A with a multiplication morphism m : A⊗ A → A and a unit morphism u : k → A satisfying the associativity and the unitary conditions of usual associative algebras but expressed in diagrams of the category Vec Φ G . More precisely, an algebra A in Vec Φ G is a finite-dimensional G-graded space ⊕ g∈G A g with a multiplication · such that for all homogeneous elements a, b, c ∈ A. Here and below, |a| denotes the G-degree of a. There is also a unit element 1 in A such that for all a ∈ A. Further, if the Gr-category Vec Φ G is braided with braiding R, then we say that two homogeneous elements a, b ∈ A are commutative if If any two homogeneous elements commute, then we say that A is a commutative algebra in Vec Φ G , R . Dually, a coalgebra in the linear Gr-category Vec Φ G is an object C with a comultiplication morphism ∆ : C → C ⊗ C and a counit morphism ε : C → k satisfying the coassociativity and counitary axioms in the category Vec Φ G . We remark that algebras in Vec Φ G are also called G-graded quasialgebras in [1]. In addition, morphisms for algebras and coalgebras in Vec Φ G can be defined in an obvious manner and thus we omit further details.

Twisted group algebras
Algebras in Vec Φ G 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 : G × G → k * with F (e, x) = F (x, e) = 1. Then we can define a twisted group algebra k F [G] which has the same vector space as the group algebra k[G] but a different product twisted by F , namely Clearly, k F [G] is an algebra in the linear Gr-category Vec ∂F is the differential of F . Note that, the twisted group algebra k F [G] is associative in the usual sense if and only if F is a 2-cocycle, i.e., ∂F = 0, on G. Furthermore, if G is abelian then Vec ∂F G is symmetric with braiding given by and k F [G] is commutative in the symmetric Gr-category Vec ∂F G , R F . Many interesting algebras appear as twisted group algebras. Here we recall some examples presented in [1,2,15]. Let R denote the field of real numbers, Z 2 = {0, 1} the cyclic group of order 2, and Z n 2 the direct product of n copies of Z 2 . Elements of Z n 2 are written as n-tuples of {0, 1} and the group product is written as Then the associated twisted group algebra R F Cl [Z n 2 ] is the well-known real Clifford algebra Cl 0,n , see [2] for detail. This recovers the algebra of complex numbers C when n = 1 and the algebra of quaternions H when n = 2. Note that Cl 0,n is associative in the usual sense since the function F Cl is a 2-cocycle.
Then the twisted group algebra R F O [Z n 2 ] is the algebra of higher octonions 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 O n are nonassociative in the usual sense as the function F O is not a 2-cocycle.

Braided tensor products
If A 1 and A 2 are two algebras in a symmetric linear Gr-category Vec Φ G , R , then there is a natural braided tensor product algebra A 1 ⊗A 2 living in same category with product given by 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 Vec Φ G , R , then there is a braided tensor product coalgebra C 1 ⊗C 2 in the same category with coproduct defined by for all homogeneous a ∈ C 1 , b ∈ C 2 . Here and in the following we use Sweedler's sigma notation ∆(a) = a 1 ⊗ a 2 for coproduct.

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 Vec Φ G , R .
Note that a weak Hopf algebra in Vec Φ G , R is a Hopf algebra in Vec Φ G , R if and only if the comultiplication is unit-preserving, and if and only if the counit is a morphism of algebras in

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 Φ a 3-cocycle on G. It is well known that k[G] has a natural Hopf algebra structure. If we extend Φ trilinearly to (k[G]) ⊗3 , then we have a dual quasi-Hopf algebra (k[G], Φ) where Φ is an associator, see [1]. In this terminology, the linear Gr-category Vec Φ G is exactly the comodule category of (k[G], Φ). By definition, a gauge transformation, or a twisting, on the dual quasi-Hopf algebra (kG, Φ) is a convolution-invertible linear In other words, F is a bilinear expansion of a normalized 2-cochain on G. Given a gauge transformation F on Q = (k[G], Φ), one can define a new dual quasi-Hopf algebra Q F = (k[G], Φ∂F ), see [1] for more details. The corresponding linear Gr-category of (k[G], Φ F ) is Vec Φ∂F G , which is tensor equivalent to Vec Φ G . The tensor equivalence is given by (F, ϕ 0 , ϕ 2 ) : for all U, V and u ∈ U x , v ∈ V y . The tensor equivalence F also induces a one-to-one correspondence between the set of braidings of Vec Φ G and that of Vec Φ∂F G . Given a braiding R of Vec Φ G , define F(R) by Then F(R) is a braiding of Vec Φ∂F 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 Vec Φ G , then A F := F(A) is an algebra in Vec Φ∂F G . Note that the multiplication of A F is given by for all homogeneous u ∈ C. Moreover, if H is a (weak) Hopf algebra in Vec Φ G , then H F := F(H) is a (weak) Hopf algebra in Vec Φ∂F G , etc. The merit of these gauge transformations lies in that, studying an algebra A in Vec Φ G amounts to studying the algebra A F in Vec Φ∂F G . By choosing suitable twisting F , there might be a chance to gauge transform an algebra A in Vec Φ G to a simpler algebra A F in Vec Φ∂F G .

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.

Generalized Clif ford 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 , . . . , q m ∈ k * . By definition, the generalized Clifford algebra C (n) (q 1 , q 2 , . . . , q m ) is the associative k-algebra with unit 1 generated by e 1 , e 2 , . . . , e m subject to the relations e n i = q i 1, ∀ i and e i e j = ωe j e i , ∀ i > j.

Formulas of 2-cocycles
By Z m n we denote the group Z n × · · · × Z n m and by [a] the integer part of a rational number a.
Let 0 ≤ x ≤ n − 1 denote the residue of the integer x divided by n. Elements g ∈ Z m n are written in the form (g 1 , g 2 , . . . , g m ) and the product of Z m n is written as +.
Take a primitive n-th root of unity ω and fix q 1 , . . . , q m ∈ k * . Define F : Z m n × Z m n → k * by for all g, h ∈ Z m n . It is easy to see that F (g, 0) = F (0, g) = 1, and that Hence F GCA is a 2-cocycle on Z m n . Define

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 , . . . , q m ) as a twisted group algebra.
Consequently, the latter is an algebra in the symmetric Gr-category (Vec Z m n , R GCA ). Proof . It is easy to see that C (n) (q 1 , q 2 , . . . , q m ) has a basis e g 1 1 e g 2 2 · · · e gm m | g i ∈ Z n , i = 1, 2, . . . , m . Define a map Clearly ψ is a linear isomorphism, so it remains to verify that ψ is also multiplicative. By direct computation, we have Hence we have proved that ψ is an isomorphism of algebras.

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. Note that the braiding o is trivial, hence the braided tensor product in (Vec Z m n , o) is the usual tensor product and so we omit the "hats" in the preceding equation. Now by applying F to (3.3), the claimed isomorphisms follow.

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 Recall that the elements of Z n are denoted byī, 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 The coassociativity of ∆ implies that δ(ī, j + k)δ(j,k) = δ(ī,j)δ(i + j,k), ∀ī,j,k.

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  With this, we may discover the categorical weak Hopf structures in suitable symmetric Grcategories of some well known algebras by properly specifying the twisted functions. In particular, if we take F = F GCA , then we determine the categorical weak Hopf structure of generalized Clifford algebras via Proposition 3.1. · · · e (gm−hm) m , ε e g 1 1 · · · e gm m = n m δ g,0 , and S e g 1 1 · · · e gm m = e g 1 1 · · · e gm m , for all e g 1 1 e g 2 2 · · · e gm m ∈ C n (q 1 , . . . , q m ) and all g = (g 1 , g 2 , . . . , g m ), h = (h 1 , . . . , h m ) ∈ Z m n , where Proof . Direct consequence of Proposition 3.1 and Corollary 4.3.

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 k be a field with char k |G|, and let F be an arbitrary 2-cochain on G. Then the twisted group algebra k F [G] has a unique weak Hopf structure in the symmetric Gr-category Vec ∂F G , R F with comultiplication ∆, counit ε and antipode S given by If we specify G = Z n 2 and F the Albuquerque-Majid cochain F CD of Cayley-Dickson algebras [1], then we determine completely the weak Hopf structures of Cayley-Dickson algebras in Vec ∂F CD Z n 2 , R F CD . This improves the main result of Bulacu [3]. If we specify G = Z n 2 and F = F O , the Morier-Genoud-Ovsienko cochain of higher octonions [15], then we determine completely the weak Hopf structures of higher octonions O n in the symmetric Gr-category Vec