Automorphisms of finitely generated relatively free bicommutative algebras
type: Item
instance of: scholarly article
title: Automorphisms of finitely generated relatively free bicommutative algebras
zbMATH Open document ID: 1498.17002
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 2021-04-29T00:00:00Z
review text: The present paper is about a study of tame and wild automorphisms of free bicommutative algebras. Each tame automorphism can be represented as a composition of elementary automorphisms and each wild automorphism is non-tame. It is known, that each automorphism of two-generated free associative and free associative commutative algebras are tame. On the other hand, there is a wild automorphism of two-generated free Leibniz algebra [\textit{A. T. Abdykhalykov} et al., Commun. Algebra 29, No. 7, 2953--2960 (2001; Zbl 0978.17001)] and three-generated associative and free associative commutative algebras [\textit{I. P. Shestakov} and \textit{U. U. Umirbaev}, J. Am. Math. Soc. 17, No. 1, 197--227 (2004; Zbl 1056.14085); \textit{U. U. Umirbaev}, J. Reine Angew. Math. 605, 165--178 (2007; Zbl 1126.16021)]. An algebra \(B\) over a field is called bicommutative if it satisfies the identities \[ x(yz) = y(xz), \ \ (xy)z = (xz)y. \] Wild automorphisms are constructed in two-generated and three-generated free bicommutative algebras. Moreover, for any \(n \geq 2,\) a wild automorphism is constructed in the \(n\)-generated free associative bicommutative algebra which is not stably tame and can not be lifted to an automorphism of the \(n\)-generated free bicommutative algebra.
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17A50
zbMATH DE Number: 7341181
zbMATH Keywords: automorphism
zbMATH Keywords: associative bicommutative algebra
zbMATH Keywords: bicommutative algebra
zbMATH Keywords: Jacobian matrix
reviewed by: I. B. Kaĭgorodov
MaRDI profile type: Publication
full work available at URL: j.jpaa.2020.106636
OpenAlex ID: W3104008991
cites work: AUTOMORPHISMS OF TWO-GENERATED FREE LEIBNIZ ALGEBRAS
cites work: Obstructions to lifting automorphisms of free algebras
cites work: Q3808246
cites work: Automorphisms of a Free Associative Algebra of Rank 2. II
cites work: A jacobian conjecture for free associative algebras
cites work: Noetherianity and Specht problem for varieties of bicommutative algebras
cites work: The strong Anick conjecture is true.
cites work: Q4918428
cites work: Q5818521
cites work: Q5622339
cites work: Combinatorial methods. Free groups, polynomials, and free algebras
cites work: Q4063564
cites work: The Nagata automorphism is wild
cites work: The tame and the wild automorphisms of polynomial rings in three variables
cites work: On the extension of automorphisms of polynomial rings
cites work: The Anick automorphism of free associative algebras
cites work: Subalgebras and automorphisms of polynomial rings.
DOI: 10.1016/J.JPAA.2020.106636
description: scientific article; zbMATH DE number 7341181