On codimension growth of finitely generated associative algebras
type: Item
instance of: scholarly article
title: On codimension growth of finitely generated associative algebras
zbMATH Open document ID: 0920.16012
published in: Advances in Mathematics
publication date: 1999-07-19T00:00:00Z
review text: Let \(A\) be a PI-algebra over a field \(F\) of characteristic 0 and let \(c_n(A)\) be the \(n\)-th codimension of (the multilinear polynomial identities of) \(A\), \(n=1,2,\ldots\). Starting in the early 70's when Regev used his exponential bounds of the codimensions to show that the tensor product of two PI-algebras is PI again, a lot of investigations have been devoted to the asymptotic behaviour of the codimension sequences both in the case of arbitrary PI-algebras and in the case of some special PI-algebras important from a different point of view. As a first approximation to the more precise estimates of the growth of the codimensions of a PI-algebra \(A\) one may study the behaviour of \(\root n\of{c_n(A)}\). A well known conjecture among the mathematicians working in PI-theory is that \(\text{Inv}(A)=\lim_{n\to\infty}\root n\of{c_n(A)}\) always exists and is an integer. In a series of two important papers [the second being \textit{A. Giambruno, M. Zaicev}, Adv. Math. 142, No. 2, 221-243 (1999; reviewed below, Zbl 0920.16013)], the authors give the affirmative answer to the above conjecture. In the paper under review they handle the case of finitely generated PI-algebras. Since the theory of Kemer gives that the polynomial identities of any finitely generated PI-algebra coincide with those of some finite dimensional algebra, it is sufficient to consider the case when \(\dim_FA<\infty\). The authors describe an effective procedure based on the Wedderburn-Malcev decomposition (\(A=B+J\), where \(J\) is the Jacobson radical of \(A\) and \(B=B_1\oplus\cdots\oplus B_n\) is the decomposition of the semisimple part of \(A\) into simple components) how to find explicitly \(\text{Inv}(A)\). Several interesting consequences of the main result are obtained as well. In particular, the finite dimensional algebra \(A\) is central simple over the base field \(F\) if and only if \(\text{Inv}(A)=\dim_FA\).
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R20
zbMATH DE Number: 1268547
zbMATH Keywords: algebras with polynomial identities
zbMATH Keywords: finitely generated algebras
zbMATH Keywords: codimensions of T-ideals
zbMATH Keywords: Wedderburn-Malcev decompositions
zbMATH Keywords: central simple algebras
author: Antonio Giambruno
author: Mikhail Zaicev
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
full work available at URL: aima.1998.1766
OpenAlex ID: W2035923919
cites work: Applications of hook Young diagrams to P.I. algebras
cites work: On the codimensions of the verbally prime P.I. algebras
cites work: Q5340151
cites work: A conjecture of Regev about the Capelli polynomial
cites work: Simple and semisimple Lie algebras and codimension growth
cites work: Q4165477
cites work: Existence of identities in \(A \otimes B\)
cites work: Algebras satisfying a Capelli identity
cites work: Asymptotic values for degrees associated with strips of Young diagrams
cites work: Codimensions and trace codimensions of matrices are asymptotically equal
DOI: 10.1006/AIMA.1998.1766
description: scientific article; zbMATH DE number 1268547