Eventually non-decreasing codimensions of *-identities
type: Item
instance of: scholarly article
title: Eventually non-decreasing codimensions of *-identities
zbMATH Open document ID: 1468.17007
published in: Archiv der Mathematik
publication date: 2021-05-14T00:00:00Z
review text: Let \(A\) be an algebra (not necessarily associative) over a field \(F\) of characteristic 0. The polynomial identities of \(A\) are completely determined by the multilinear ones. Let \(F\langle X\rangle\) be the free associative algebra over \(F\) that is freely generated by the set of variables \(X=\{x_1,x_2, \ldots\}\), and denote by \(P_n\) its vector subspace of all multilinear polynomials in \(x_1\), \dots, \(x_n\). If \(\operatorname{Id}(A)\) is the T-ideal of \(A\) then one can study \(P_n\cap \operatorname{Id}(A)\), \(n\ge 1\), in order to know \(\operatorname{Id}(A)\). As \(P_n\) is a left module over the symmetric group \(S_n\), and \(\operatorname{Id}(A)\) is closed under permutations of the variables, one has that \(P_n\cap \operatorname{Id}(A)\) is a submodule. Hence the representation theory of \(S_n\) can be employed in the study of PI algebras. It turns out it is much more convenient to study the factor module \(P_n/(P_n\cap \operatorname{Id}(A))\). Its dimension \(c_n(A)\) is the \(n\)th codimension of \(A\) (of \(\operatorname{Id}(A)\), more precisely). The codimension sequence of a PI algebra was introduced and first studied by \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)]. Regev proved that if \(A\) satisfies an identity of degree \(d\) then \(c_n(A)\le (d-1)^{2n}\) for every \(n\), and applied this fact in order to prove that the tensor product of PI algebras is PI as well. The fact that the growth of \(c_n(A)\) is bounded by an exponential function implies that \(P_n\cap \operatorname{Id}(A)\) becomes very large if \(\operatorname{Id}(A)\ne 0\). Furthermore it was proved by \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 140, No. 2, 145--155 (1998; Zbl 0920.16012); Adv. Math. 142, No. 2, 221--243 (1999; Zbl 0920.16013)], that \(\lim_{n\to\infty} (c_n(A)^{1/n})\) always exists and is a nonnegative integer, called the PI exponent of \(A\), \(\exp(A)\). It is an important problem to determine more precisely the gr…
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 7348117
zbMATH Keywords: identities with involution
zbMATH Keywords: codimensions
zbMATH Keywords: exponential growth
author: Ivan Shestakov
author: Mikhail Zaicev
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: s00013-020-01567-9
OpenAlex ID: W3120838864
cites work: Properties of hook Schur functions with applications to P. I. algebras
cites work: Codimensions of star-algebras and low exponential growth
cites work: Q5714236
cites work: Growth of polynomial identities: is the sequence of codimensions eventually non-decreasing?
DOI: 10.1007/S00013-020-01567-9
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description: scientific article; zbMATH DE number 7348117