Colength of varieties of linear algebras
type: Item
instance of: scholarly article
title: Colength of varieties of linear algebras
zbMATH Open document ID: 1114.17001
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: Let \textbf{V} be a variety of (not necessarily associative) algebras over a field \(\Phi\) of characteristic 0. The \(n\)-th cocharacter \(\chi_{n}({\mathbf V})\), \(n=1,2,\ldots\), is defined as in the case of associative PI-algebras, as the character of the symmetric group \(S_{n}\) acting on the multilinear polynomials of the absolutely free algebra modulo the polynomial identities of \textbf{V}. The colength \(l_{n}({\mathbf V})\) is defined as the sum \(\sum_{\lambda\vdash n}m_{\lambda}\) of the multiplicities \(m_{\lambda}\) in the decomposition \(\chi_{n}({\mathbf V})=\sum_{\lambda\vdash n}m_{\lambda}\chi_{\lambda}\), where \(\chi_{\lambda}\) is the irreducible \(S_{n}\)-character related to the partition \(\lambda\) of \(n\). In the case of associative PI-algebras, the colength \(l_{n}({\mathbf V})\) is polynomially bounded. In the paper under review the authors construct a family of nonassociative algebras which satisfy the fourth Capelli identity and generate varieties \textbf{V} with exponentially growing colength. The construction of the algebras is quite interesting and it seems that it may be used (or at least modified) to build other examples in the theory of nonassociative algebras with polynomial identity.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17A50
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 5155477
zbMATH Keywords: nonassociative algebra
zbMATH Keywords: variety of algebras
zbMATH Keywords: Capelli identities
zbMATH Keywords: colength
zbMATH Keywords: codimensions
zbMATH Keywords: polynomial growth
zbMATH Keywords: exponential growth
author: Mikhail Zaicev
author: Sergey Mishchenko
MaRDI profile type: Publication
full work available at URL: s11006-006-0056-0
OpenAlex ID: W1971145679
cites work: Applications of hook Young diagrams to P.I. algebras
cites work: Asymptotic behaviour of the colength growth functions of varieties of Lie algebras
cites work: Q4526849
cites work: ON THE COLENGTH OF A VARIETY OF LIE ALGEBRAS
cites work: Q4268148
cites work: Algebras with intermediate growth of the codimensions.
DOI: 10.1007/S11006-006-0056-0
description: scientific article; zbMATH DE number 5155477