Deformations of nonassociative algebras and integrable differential equations
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.
systems of partial differential equations
scientific article; zbMATH DE number 824537