Asymptotic properties of free finitely generated algebras of certain varieties
type: Item
instance of: scholarly article
title: Asymptotic properties of free finitely generated algebras of certain varieties
zbMATH Open document ID: 0547.17008
DOI: 10.1007/BF01978875
published in: Algebra and Logic
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187246
review text: The author discusses the rate of growth of finitely generated algebras of certain varieties of not necessarily associative algebras. An important characteristic of such a growth is the number \(\tau\) (d) such that \(c_ 1 n^{\tau (d)}<\dim F(n)<c_ 2 n^{\tau (d)}\) (for sufficiently large n) where F(n) is the homogeneous component of degree n in a d-generated varietally free algebra. The existence of \(\tau\) (d) is shown here for the varieties generated by the algebra of \(r\times r\) matrices over a field k as well as by the Cayley-Dixon algebra over k and some related algebras. In fact this number is equal to the transcendence degree of the center of the varietally free generating algebra of the corresponding variety. The additivity of the above characteristic is shown for certain special products of algebras. This property proves useful in computing the rate of growth of free algebras of some not necessarily associative varieties.
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 3871602
zbMATH Keywords: variety of algebras
zbMATH Keywords: alternative algebras
zbMATH Keywords: growth of finitely generated algebras
zbMATH Keywords: homogeneous component
zbMATH Keywords: varietally free algebra
zbMATH Keywords: free algebras
author: Aleksandr V. Grishin
MaRDI profile type: Publication
cites work: Q3889234
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OpenAlex ID: W1967992216
description: scientific article; zbMATH DE number 3871602