The ``measure of non-Lieness'' for Mal'tsev algebras
type: Item
instance of: scholarly article
title: The ``measure of non-Lieness'' for Mal'tsev algebras
zbMATH Open document ID: 0792.17024
DOI: 10.1007/BF02259852
author: V. T. Filippov
published in: Algebra and Logic
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/187608
review text: Let \(\Phi\) be a commutative-associative ring with 1/6, and \(A\) be a Mal'tsev \(\Phi\)-algebra. If \(A\) can be isomorphically embedded in the commutator algebra \(B^{(-)}\) of some alternative algebra \(B\), then \(A\) is said to be special, and \(B\) is called an alternative enveloping algebra for \(A\). Let \(J(A)\) denote the ideal generated in \(A\) by all Jacobians \(J(x,y,z)= (xy)z+ (yz)x+ (zx)y\), where \(x,y,z\in A\). The author proves that \(J(A)\) is special; and if \(\Phi\) is a field such that \(J(A)\) is finite-dimensional over \(\Phi\), then an alternative enveloping algebra for \(J(A)\) is also finite-dimensional. The proof of this result involves the construction of an invariant symmetric bilinear form on \(J(A)\) over a commutative subalgebra \(K\) of the centroid of \(J(A)\). It is also proved that \(J(A)\) is Shirshov locally-finite over \(K\).
Mathematics Subject Classification ID: 17D10
zbMATH DE Number: 536661
zbMATH Keywords: special Malcev algebra
zbMATH Keywords: special ideal
zbMATH Keywords: Shirshov locally-finite ideal
zbMATH Keywords: alternative enveloping algebra
zbMATH Keywords: symmetric bilinear form
MaRDI profile type: Publication
cites work: On Engelian Mal'tsev algebras
cites work: Prime Mal'tsev algebras
cites work: Central simple Mal'tsev algebras
cites work: Varieties of special algebras
cites work: Imbedding of Mal'tsev algebras into alternative algebras
cites work: Subrings of invariants of the finite group of automorphisms of a Jordan ring
cites work: Malcev Algebras
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description: scientific article; zbMATH DE number 536661