Central polynomials of algebras and their growth
type: Item
instance of: scholarly article
title: Central polynomials of algebras and their growth
zbMATH Open document ID: 1498.16026
DOI: 10.1007/978-3-030-63111-6_10
publication date: 2021-12-15T00:00:00Z
review text: A central polynomial for an algebra \(A\) is a polynomial \(f(x_1,\ldots, x_n)\) such that \(f(a_1,\ldots,a_n)\) lies in the center of \(A\) for all \(a_1,\ldots,a_n\in A\). This paper studies two codimension sequences for central polynomials analoguous to the ordinary codimension sequence. Let \(P_n\) be the space of multilinear degree \(n\) polynomials in \(x_1,\ldots,x_n\). Then let \(I_n\) be the space of polynomial identities of \(A\) contained in \(P_n\) and let \(I_n\subset \Delta_n\) be the space of central polynomials. The three codimensions are \(c_n=\dim (P_n/I_n)\), the ordinary codimension; \(c_n^Z=\dim(P_n/J_n)\), the central codimension; and \(\delta_n=\dim(\Delta_n/I_n)\), the proper central codimension. Note that \(c_n^Z+\delta_n=c_n\). These codimensions were defined by \textit{A. Regev} [Commun. Algebra 44, No. 10, 4411--4421 (2016; Zbl 1354.16023)] and studied by the authors in [Proc. Am. Math. Soc. 147, No. 3, 909--919 (2019; Zbl 1409.16018)]. All three codimension sequences have an integral exponential rate of growth, namely for each one the limit of the \(n\)-th root of the \(n\)-th codimension converges to an integer, and the ordinary and central codimensions have the same exponential rates of growth. The current paper describes the results of these papers with some interesting examples and some new results. These include algebras with arbitrarily large gaps between the exponential rates of growth of \(c_n(A)\) and \(\delta_n(A)\) and a non-associative algebra in which all three exponential rates of growth are non-integral. For the entire collection see [Zbl 1461.16003].
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 7445612
zbMATH Keywords: central polynomial
zbMATH Keywords: polynomial identity
zbMATH Keywords: codimension
zbMATH Keywords: exponential growth
author: Antonio Giambruno
author: Mikhail Zaicev
reviewed by: Allan Berele
MaRDI profile type: Publication
full work available at URL: 978-3-030-63111-6 10
OpenAlex ID: W3139091345
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description: scientific article; zbMATH DE number 7445612