Regular Gradings on Algebras

Define the Grassmann algebra $E$ to be the free noncommutative algebra $F<e_i \mid i\in \mathbb{N}>$ modulo the relations $e_i ^2=0$ and $e_i e_j = - e_j e_i$ for all $i \neq j$ . Setting $E_1 = span \{e_{i_1}\cdots e_{i_{2n}} \mid \; i_1 <\cdots <i_{2n} \}$ and $E_{-1} = span \{e_{i_1}\cdots e_{i_{2n+1}} \mid \; i_1 <\cdots <i_{2n+1} \}$ , one easily checks that $E=E_1 \oplus E_2$ is a $C_2$ -grading of the Grassmann algebra.

Clearly, the Grassmann algebra is not commutative, yet it is close to being commutative. Using the relations defining the Grassmann algebra, we get that

$(e_{i_1} \cdots e_{i_n})$ $(e_{j_1} \cdots e_{j_m})$ $= (-1)^{nm}(e_{j_1}\cdots e_{j_m})(e_{i_1}\cdots e_{i_n})$ .

Extending this linearly to the homogeneous components, we get that for any $a,b \in E$ homogeneous we have $ab=\tau(\deg(a),\deg(b))ba$ where $\tau(1,1)=\tau(1,-1)=\tau(-1,1)=1$ and $\tau(-1,-1)=1$ . In other words, homogeneous elements commute up to a scalar.

This notion was generalized by Regev and Seeman [1] with the following definition: Let G be a finite abelian group. A G-gradingon an algebra $A$ is said to be regular if there is a commutation function $\theta:G\times G \to F^\times$ such that

For any tuple $g_1,\;g_2,\cdots,\;g_n \in G$ there are $a_i \in A_{g_i}$ such that $\prod_1 ^n a_i \neq 0$ .
For each $g,h\in G$ and any $a_g \in A_g,\;b_h \in A_h$ we have $a_g b_h = \theta(g,h) b_h a_g$ .

Other examples for such regular gradings come from twisted group algebras. The commutation function is defined by $\theta(g,h)=\frac{\alpha(g,h)}{\alpha(h,g)}$ , and the first condition in the definition is satisfied since all the nonzero homogeneous elements are invertible. In addition, every matrix algebra is isomorphic to some twisted group algebra, so every matrix algebra admits some regular grading.

Let $A$ be some regularly G-graded algebra. Setting $H=\{h\in G\;\mid\; \theta(h,g)=1\}$ , we get that the induced $G/H$ grading on $A$ is still regular and satisfies that for all $\bar e \neq \bar h \in G/H$ there is some $bar h '\in G/H$ such that $\bar \theta (\bar h , \bar h ')\neq 1$ . A grading which satisfies this condition is called minimal. An algebra $A$ may admit several minimal regular gradings with non isomorphic groups. It was conjectured by Bahturin and Regev [2] that also these groups are not isomorphic, they do have the same cardinality.

Our results [3]:

As the definition of a regular grading is actually a condition on the ideal of identities, we give a classification up to PI-equivalence of all regular graded algebras. In particular, up to a PI-equivalence, every regular graded algebra is a combination of a twisted group algebra and the Grassmann algebra.

Extending the definition to arbitrary groups, we use the classification above and show that the cardinality is indeed an invariant of the algebra. In particular, this invariant is called the exponent of the algebra and appears in PI-theory. The exponent measures in a sense how many multilinear identities of degree n the algebra has when n goes to infinity.

Questions:

The classification of all abelian groups G, such that there are algebras $A$ with minimal regular G-grading, is well known. Every such group must either be $G=H\times H$ or $G=H\times H\times C_2$ for some abelian group H (the $C_2$ part comes from the Grassmann algebra).

A group G which has a 2-cocycle $\alpha$ such that $F^\alpha G \cong M_r(k)$ is called a central type. The classification of non abelian central type groups is still an open question. By the extended definition every central type group admits a minimal regular grading on some algebra $A$ .

Is there a classification of groups admitting minimal regular gradings, if we are given the classifiction of groups of central type?

Remarks:

A similar process was done with Lie algebras. A G-graded Lie algebra is called a color Lie algebra if $[a,b] = -\theta (\deg(a),\deg(b)) [b,a]$ . It appears that the function $theta$ in this case are exactly the commutation functions appearing in the regular grading setting. These “commutation function” also appear (with similar definitions) in Hopf Algebras.

Bibliography:

A. Regev and T. Seeman, $\mathbb{Z}_2$ -Graded tensor products of PI-algebras, Journal of Algebra, Volume 291(1):(2005), 274-296.
DOI: 10.1016/j.jalgebra.2005.01.049.
A. Regev and Y. Bahturin, Graded tensor products, Journal of Pure and Applied Algebra 213(9): (2009), 1643-1650.
DOI 10.1016/j.jpaa.2008.12.010‏
E. Aljadeff and O. David E. Aljadeff, On regular $G$ -gradings, to appear in Transactions of the American Mathematical Society.
arXiv: arXiv:1212.0343v2 [math.RA]‏