Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2
Classification of small dimensional algebras from a certain variety defined by some polynomial identities is one of the classical problems in the theory of non-associative algebras. There are two principal types of classifications: algebraic and geometric classifications. The notion of Mal'cev algebras appeared in 1955 in a paper of a famous Russian mathematician \textit{A. Mal'cev} [Mat. Sb., N. Ser. 36(78), 569--576 (1955; Zbl 0065.00702)]. The variety of Mal'cev algebras is a generalization of Lie algebras. Mal'cev algebras have many relations with alternative algebras and Moufang loops. On the other hand, it is a particular case of binary Lie algebras. The first algebraic classification of $6$-dimensional nilpotent non-Lie Mal'cev was given over an arbitrary field of characteristic zero in one paper of \textit{E. N. Kuz'min} [Algebra Logika 9, 691--700 (1970; Zbl 0244.17018)]. Based on this algebraic classification, the geometric classification of $6$-dimensional complex nilpotent Mal'cev algebras was given in [the reviewer et al., Commun. Algebra 46, No. 11, 4928--4940 (2018; Zbl 1441.17027)]. The present paper is devoted to the algebraic classification of $6$-dimensional nilpotent Mal'cev algebras over an arbitrary field of characteristic not $2$. As follows from this result, all $6$-dimensional nilpotent non-Lie Mal'cev algebras are metabelian (i.e. they are satisfying the identity (xy)(zt)=0). Note that, there are some $6$-dimensional non-metabelian nilpotent Lie algebras. It was a reason for the second part of the paper, where the algebraic classification of $7$-dimensional nilpotent non-metabelian non-Lie Mal'cev algebras over an arbitrary field of characteristic not $2$ was obtained.
scientific article; zbMATH DE number 6839336
scientific article