Tensor extensions of Lax equations

Abstract

The Lax equations dL/dt = [M,L] play an important role in the integrability theory of nonlinear evolution equations and quantum dynamics. In this work, tensor extensions of the Lax equations are suggested with M : V → V and L : Tk(V ) → V , k = 1, 2, . . ., on a complex vector space V . These extensions belong to the generalised class of Lax equations (introduced earlier by Bordemann) dL/dt = ρk(M)L where ρk is a representation of a Lie algebra. The case k = 1, ρ1 = ad corresponds to the usual Lax equations. The extended Lax pairs are studied from the point of view of isomorphic deformations of multilinear structures, conservation laws, exterior algebras and cochain symmetries.