On group gradings on PI-algebras (original) (raw)
Related papers
2012
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.
Transactions of the American Mathematical Society, 2014
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.
Group graded PI-algebras and their codimension growth
Israel Journal of Mathematics, 2012
Let W be an associative PI-algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(We) denote the codimension growth of W and of the identity component We, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W) ≤ |G| 2 exp(We). The inequality is known in case the algebra W is affine (i.e. finitely generated). Here we prove the conjecture in general.
Group Gradings on Associative Algebras
Journal of Algebra, 2001
Rg be a G-graded ring. We descnoe all types of gradings on R if G is torsion free and R is Artinian semisimpIe. If R is a matrix algebra over an algebraically closed field F, then we give a description of all G-gradings on R provided that G is an abelian group. In the case of an abelian group G we also classify all finite-dimensional graded simple algebras and finite-dimensional graded division algebras over an algebraically closed field of characteristic zero.
Group-graded algebras with polynomial identity
Israel Journal of Mathematics, 1998
Let G be a finite group and let R -~geG Rg be any associative algebra over a field such that the subspaces Rg satisfy RgRh C Rgh. We prove that if R1 satisfies a PI of degree d, then R satisfies a PI of degree bounded by an explicit function of d and the order of G. This result implies the following: if H is a finite-dimensional semisimple commutative Hopfalgebra and R is any H-module algebra with R g satisfying a PI of degree d, then R satisfies a PI of degree bounded by an explicit function of d and the dimension of H.
Group Gradings on Filiform Lie Algebras
Communications in Algebra, 2015
We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial Z k-gradings.
Communications in Algebra, 2009
In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G2 over an algebraically closed field F of characteristic zero.
Group graded algebras and multiplicities bounded by a constant
Journal of Pure and Applied Algebra, 2013
Let G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a PI-algebra, the graded codimensions of A are exponentially bounded and one can study the corresponding graded cocharacters via the representation theory of products of symmetric groups. Here we characterize in two different ways when the corresponding multiplicities are bounded by a constant.
On the codimension growth of GGG-graded algebras
Proceedings of the American Mathematical Society, 2010
Let W W be an associative PI-affine algebra over a field F F of characteristic zero. Suppose W W is G G -graded where G G is a finite group. Let exp ( W ) \exp (W) and exp ( W e ) \exp (W_{e}) denote the codimension growth of W W and of the identity component W e W_{e} , respectively. We prove exp ( W ) ≤ | G | 2 exp ( W e ) . \exp (W)\leq |G|^2 \exp (W_{e}). This inequality had been conjectured by Bahturin and Zaicev.