Recent zbMATH articles by "Mishchenko, Sergeĭ Petrovich"https://zbmath.org/atom/ai/mishchenko.sergei-petrovich2024-03-13T18:33:02.981707ZWerkzeugNilpotent varieties and metabelian varietieshttps://zbmath.org/1528.170042024-03-13T18:33:02.981707Z"Valenti, Angela"https://zbmath.org/authors/?q=ai:valenti.angela"Mishchenko, Sergey"https://zbmath.org/authors/?q=ai:mishchenko.sergei-petrovichThe authors study varieties of nonassociative algebras having polynomial growth of codimensions over a field of characteristic zero. They cite known results when the codimension growth is polynomial in case of associative, Lie and Leibnitz algebras. Some results on the codimension growth are also known for Jordan algebras. Next they describe their recent results in the class of left nilpotent algebras of index two, i.e. algebras satisfying the identity \(x(yz)\equiv 0\), see [\textit{S. Mishchenko} and \textit{A. Valenti}, J. Algebra 518, 321--342 (2019; Zbl 1459.17002)]. Recently the authors constructed a correspondence between left nilpotent algebras of index two and commutative metabelian algebras or anticommutative metabelian algebras (both are considered in the class of absolutely free algebras) and proved that the codimensions sequences of the corresponding algebras coincide up to a constant. [\textit{S. P. Mishchenko} and \textit{A. Valenti}, J. Pure Appl. Algebra 225, No. 3, Article ID 106538, 9 p. (2021; Zbl 1472.17008)]. Using these results, they transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.
Reviewer: Victor Petrogradsky (Brasília)