Automorphisms of finitely generated relatively free bicommutative algebras. (English) Zbl 1498.17002
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 [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 [I. P. Shestakov and U. U. Umirbaev, J. Am. Math. Soc. 17, No. 1, 197–227 (2004; Zbl 1056.14085); 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.
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.
Reviewer: Ivan Kaygorodov (Covilhã)
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