Group graded algebras and multiplicities bounded by a constant (original) (raw)

Cocharacters of group graded algebras and multiplicities bounded by one

Linear and Multilinear Algebra, 2017

Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the T G-ideals Id G (A) of graded identities of A such that the multiplicities m λ in the graded cocharacter of A are bounded by one. We do so by exhibiting a set of identities of the T G-ideal. As a consequence we characterize the varieties of G-graded algebras whose lattice of subvarieties is distributive.

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.

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.

An upper bound for the index of multiplicities in the cocharacters of PI-algebras

Journal of Mathematical Sciences, 2007

We give an upper bound for the index of the polynomial degree limiting the multiplicities in the cocharacter of a variety of associative algebras over a field of characteristic zero. It was proved in [5, 6] that exp(A) always exists and is an integer. According to A. R. Kemer's results [9, Theorem 2.3], any variety of associative algebras over a field of characteristic zero can be generated by the Grassmann hull of some finite-dimensional superalgebra. This means that we can assume that A = G(C) = C (0) ⊗ G 0 + C (1) ⊗ G 1 , where G = G 0 + G 1 is the Grassmann algebra of countable rank with the natural Z 2-grading (G 0 and G 1 are the subspaces generated by all monomials of even and odd length in generators, respectively) and C = C (0) + C (1) is a finite-dimensional Z 2-graded algebra. Moreover, we can assume that C = B + J, where B is a semisimple finite-dimensional algebra,

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.

ASYMPTOTICS FOR THE MULTIPLICITIES IN THE COCHARACTERS OF SOME PI-ALGEBRAS

2000

We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some significant classes of algebras. We also give a char- acterization of finitely generated algebras for which this behavior is linear or quadratic.

Semigroup graded algebras and graded PI-exponent

Israel Journal of Mathematics, 2017

Let S be a finite semigroup and let A be a finite dimensional S-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions c S n (A) of A, i.e lim n→∞ n c S n (A). For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an S-graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals dim(A). In light of the previous, we also handle the problem of classification of all S-graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of the general problem.

Identities of graded algebras and codimension growth

Transactions of the American Mathematical Society

Let A = ⊕ g ∈ G A g A=\oplus _{g\in G}A_g be a G G -graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component A e A_e to that of the whole of A A , in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where A A is finite dimensional and A e A_e has polynomial growth.

On forms of G-graded simple algebras

2021

Let G be finite group and D a finite dimensional G-graded division algebra e-central over k (k consists of the central e-homogeneous elements of D). Restricting scalars to the algebraic closure F = k̄, we obtain a finite dimensional G-graded simple algebra. In this lecture we consider the problem in the opposite direction, namely if A is a finite dimensional G-graded simple algebra over F (with char(F ) = 0), then we ask under which conditions it admits a G-graded division algebra form (in the sense of descent theory)? (i.e. nonzero homogeneous elements are invertible). More restrictive, we ask under which conditions A admits a division algebraG-graded form? (i.e. nonzero elements are invertible). We provide a complete answer for the first question and only a partial one for the second. The main tools come from PI-theory. These allow us to construct the corresponding generic objects. Joint works with (1) Haile and Karasik, (2) Karasik.