On regular 𝐺-gradings

Abstract

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular Ggrading on A, namely a grading A = g∈G Ag that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ G n , there are elements, ai ∈ Ag i , i = 1, . . . , n, such that n 1 ai = 0 (2) for every g, h ∈ G and for every ag ∈ Ag, b h ∈ A h , we have agb h = θ g,h b h ag. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

Identities and central polynomials for real graded division algebras

“…This is a commutative (g) m -graded division algebra. We remark that D(2, −1) is weakly isomorphic to C (2) .…”

“…We denote this graded division algebra by H (4) . A coarsening of this grading is S e = 1, i , S a = j, k , this is a grading by the group H = (a) 2 ∼ = Z 2 on S = H. This grading will be denoted by H (2) . is S e = I, C , S a = ωA, ωB , S a 2 = iI, iC , S a 3 = iωA, iωB , this grading will be denoted by M 2 (C, Z 4 ).…”

“…Therefore a basis for the graded central polynomials of finite dimensional simple real algebras with a regular division grading are obtained from Corollary 1. and M 2 (C, Z 4 ) is known. Such a basis is known for the algebras H, H (2) and M Remark 6. It is well known that the algebras H and M 2 (R) satisfy the same polynomial identities, hence the previous theorems provide basis for the polynomial identities and central polynomials for H.…”

“…It was proved in [4] that H (2) and M (2) 2 satisfy the same graded polynomial identities as R = M 2 (R) with the (a) 2 -grading…”

“…In this case β(g, h) := σ(g, h)σ(h, g) −1 is a skew-symmetric bicharacter, conversely every skew-symmetric bicharacter is obtained from a 2-cocycle in this way (see [19], [2,Remark 13]). Denote C σ G the (complex) twisted group algebra, that is, the complex vector space with basis G and multiplication such that g · h := σ(g, h)gh.…”