Polynomial Identities on Superalgebras and Almost Polynomial Growth

Giambruno, Antonio; Mishchenko, S. P.; Zaicev, Mikhail

doi:10.1081/agb-100105975

Cited by 53 publications

(41 citation statements)

References 4 publications

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“…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: 98%

“…. , 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%

“…Here J(B i ) is as usual the Jacobson radical of B i . Further detailed descriptions of the varieties of G-graded algebras with polynomial growth of their graded codimensions were given in [13,33,25], see Theorem 1.3 below.…”

Section: Introductionmentioning

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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Section: Polynomial Codimension Growth and Colengthsmentioning

confidence: 98%

“…. , 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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“…For example, the structure of the Z 2 -graded identities of E with respect to its natural Z 2 -grading is well known, see for instance [9]. Recently, Di Vincenzo and da Silva gave in [6] a complete description of the Z 2 -graded polynomial identities of E with respect to any Z 2 -grading such that the generating space is Z 2 -homogeneous.…”

Section: Introductionmentioning

confidence: 99%

The $G$-graded identities of the Grassmann Algebra

Centrone¹

2016

Arch. Math. (Brno)

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“…Theorem 4.1. (Giambruno, Mischenko, Zaicev [19]) The T Z2 -graded ideal of E endowed with its natural Z 2 -grading is generated by the polynomials…”

Section: Z 2 -Graded Identitiesmentioning

confidence: 99%