Group graded algebras and multiplicities bounded by a constant

Cirrito, Alessio; Giambruno, Antonio

doi:10.1016/j.jpaa.2012.06.005

Cited by 8 publications

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References 17 publications

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“…. , n s )-cocharacter of the algebras UT g 2 , E, E a were computed (see [4,13,28,29]) and it turned out that the corresponding sequences of colengths are not bounded by any constant.…”

Section: Polynomial Codimension Growth and Colengthsmentioning

confidence: 99%

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Graded algebras with polynomial growth of their codimensions

Koshlukov

Mattina

2015

Journal of Algebra

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Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with invo-lution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group. §1. Introduction Let F be a field of characteristic zero and F X = F x 1 , x 2 ,. .. the free algebra of countable rank over F. If A is a PI-algebra over F , that is, an algebra satisfying a polynomial identity, we let Id(A) be the T-ideal of F X of identities of A. It is well known that Id(A) is completely determined by the multilinear polynomials it contains; if V n = Span{x σ(1) · · · x σ(n) | σ ∈ S n } is the space of multilinear poly-nomials in x 1 ,. .. , x n , then the sequence c n (A) = dim F Vn Vn∩Id(A) , n = 1, 2,. .. , is called the sequence of codimensions of A and it is an important numerical invariant of Id(A). It was proved by Regev in [R] that for any PI-algebra A, c n (A) is exponentially bounded, i.e., there exist constants a, α > 0 such that c n (A) ≤ aα n for all n. In this paper we study algebras A whose codimension sequence is polynomially bounded i.e., such that for all n, c n (A) ≤ an t for some constants a, t. Kemer in [K1] gave a characterization of such T-ideals in the language of the representation theory of S n. It also follows from [K2] that if c n (A) is polynomially bounded, then Id(A) = Id(B) for a suitable finite dimensional algebra B. For any finite dimensional algebra A over an algebraically closed field we shall prove that A has polynomial growth of the codimensions if and only if A = A 0 ⊕A 1 ⊕ · · · ⊕ A m where A 0 , A 1 ,. .. , A m are F-algebras such that 1) for i = 1,. .. , m, A i = B i + J i , where B i ∼ = F and J i is a nilpotent ideal of A i , 2) A 0 , J 1 ,. .. , J m are nilpotent right ideals of A and 3) A i A k = 0 for all i, k ∈ {1,. .. , m}, i = k and B i A 0 = 0.

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“…. , n s )-cocharacter of the algebras UT g 2 , E, E a were computed (see [4,13,28,29]) and it turned out that the corresponding sequences of colengths are not bounded by any constant.…”

Section: Polynomial Codimension Growth and Colengthsmentioning

confidence: 99%

“…Conversely, assume that l G n (A) ≤ k is bounded by a constant k. In this case UT g 2 , E, E a , F C h p / ∈ var G (A) for all g ∈ G, a ∈ G of order 2, and h ∈ G of order a prime p (see [4,13,28,29] and Theorem 4.1). By Theorem 1.3 this implies that var G (A) is of polynomial growth.…”

Section: Polynomial Codimension Growth and Colengthsmentioning

confidence: 99%

Graded algebras with polynomial growth of their codimensions

Koshlukov

Mattina

2015

Journal of Algebra

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show abstract

“…A similar result was obtained by Otera in [23] for finitely generated superalgebras: in this case the variety of superalgebras V does not contain the superalgebra U T 2 (trivial grading) and U T sup 2 , i.e., the algebra U T 2 with the canonical non-trivial Z 2 -grading. The latter characterization was extended in [3] for G-graded algebras, where G is any finite abelian group, by excluding from the variety of G-graded algebras V the algebra U T 2 with any G-grading. Finally, in [28], Vieira studied the same problem in the setting of finitely generated algebras with involution.…”

Section: Introductionmentioning

confidence: 99%