Deformations of nonassociative algebras and integrable differential equations
type: Item
instance of: scholarly article
title: Deformations of nonassociative algebras and integrable differential equations
zbMATH Open document ID: 0857.17001
DOI: 10.1007/BF00996121
author: S. I. Svinolupov
author: V. V. Sokolov
published in: Acta Applicandae Mathematicae
publication date: 1996-01-31T00:00:00Z
review text: The authors introduce a class of nonassociative algebras defined by an identity that requires every linear map \(K(X,Y)\) to be a derivation, where \(K(X,Y)Z:= (X,Y,Z)- (Y,X,Z)\), and \((\cdot,\cdot,\cdot)\) stands for the associator, and investigate certain deformations of these. The motivation for this lies in the investigation of certain systems of partial differential equations which, by their special shape, define an affine connection at every point. The affine connections are then subjected to a certain transformation condition that, in turn, brings the above algebras into play. (The underlying idea is to find a strategy for the production of integrable systems.) The class of algebras under consideration includes many well-known classes of nonassociative algebras, like Lie, Jordan and left-symmetric algebras. The commutative ones are precisely the Lie triple algebras investigated by H. P. Petersson and J. M. Osborn. The authors also exhibit a relation to \(\mathbb{Z}_2\)-graded Lie algebras that split into a (vector space) direct sum of subalgebras in a special way. It is shown for several of these classes that one actually obtains integrable equations, and also that the deformations (viewed on the algebra level) do not change the isomorphism class of the algebras under consideration.
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 17C99
Mathematics Subject Classification ID: 17B99
zbMATH DE Number: 824537
zbMATH Keywords: Lie algebras
zbMATH Keywords: Jordan algebras
zbMATH Keywords: nonassociative algebras
zbMATH Keywords: identity
zbMATH Keywords: systems of partial differential equations
zbMATH Keywords: affine connection
zbMATH Keywords: integrable systems
zbMATH Keywords: left-symmetric algebras
zbMATH Keywords: Lie triple algebras
reviewed by: S Walcher
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00996121
OpenAlex ID: W4248933167
description: scientific article; zbMATH DE number 824537