On generators of polynomial algebras in two commuting or non-commuting variables
type: Item
instance of: scholarly article
title: On generators of polynomial algebras in two commuting or non-commuting variables
zbMATH Open document ID: 0932.16027
DOI: 10.1016/S0022-4049(97)00134-5
author: Vladimir Shpilrain
author: Jie-Tai Yu
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-11T00:00:00Z
review text: Let \(A_2\) be the free associative algebra on \(x\), \(y\) over a field \(K\) and \(P_2\) the polynomial ring on \(x\), \(y\) over \(K\). The authors study their automorphism groups \(\Aut(A_2)\) and \(\Aut(P_2)\). They observe that these groups are isomorphic, with the Abelianization \(A_2\to P_2\) providing a natural isomorphism. They now define two subgroups of \(\Aut\). \(\Aut^0(A_2)\) is the group of augmentation-preserving automorphisms of \(A_2\) and \(\Aut_{1L}(A_2)\) generated by elementary automorphisms (i.e. of form \(x\to x+f(y)\), \(y\to y\) or \(x\to x\), \(y\to y+f(x)\)). Similar definitions apply for \(P_2\). They prove the following theorem: For any ground field \(K\), \(\Aut^0(A_2)\) can be expressed as a semidirect product of \(\Aut_{1L}(A_2)\) and \(\text{GL}_2(K)\). When the polynomials in the generators of \(\Aut_{1L}(A_2)\) have no constant or linear terms, the resultant group is a free product \(L*L\), where \(L\) is the direct sum of countably many copies of the additive group of \(K\). -- The rest of the paper is concerned with primitive elements. An algorithm is presented to distinguish primitive elements (in characteristic zero) and some necessary conditions for primitivity are given, e.g. every primitive element is reversible (admits the antiautomorphism fixing \(x\) and \(y\)).
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 1308255
zbMATH Keywords: free product algorithms
zbMATH Keywords: free associative algebras
zbMATH Keywords: polynomial rings
zbMATH Keywords: automorphism groups
zbMATH Keywords: augmentation-preserving automorphisms
zbMATH Keywords: elementary automorphisms
zbMATH Keywords: primitive elements
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308255