Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2
type: Item
instance of: scholarly article
title: Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2
zbMATH Open document ID: 1430.17102
author: Hani Abdelwahab
author: Ahmed Hegazi
author: Antonio J. Calderón Martín
published in: Algebras and Representation Theory
publication date: 2018-02-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17D10
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 6839336
zbMATH Keywords: nilpotent Maltsev algebras
zbMATH Keywords: nilpotent Lie algebras
zbMATH Keywords: automorphism group
reviewed by: I. B. Kaĭgorodov
MaRDI profile type: Publication
full work available at URL: s10468-017-9701-4
OpenAlex ID: W2621294107
cites work: 3-Filiform Leibniz algebras of maximum length, whose naturally graded algebras are Lie algebras
cites work: Leibniz algebras of nilindex \(n-3\) with characteristic sequence \((n-3,2,1)\)
cites work: \(p\)-filiform Leibniz algebras of maximum length
cites work: The classification of 4-dimensional Leibniz algebras
cites work: Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2
cites work: The classification of \(n\)-dimensional non-Lie Malcev algebras with \((n-4)\)-dimensional annihilator
cites work: Solvable Leibniz algebras with triangular nilradicals
cites work: Q5603893
cites work: Mal'tsev algebras of dimension five over a field of characteristic zero
cites work: Q4242572
cites work: Q5848903
cites work: Malcev Algebras
cites work: Q4155039
DOI: 10.1007/S10468-017-9701-4
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description: scientific article; zbMATH DE number 6839336