Тензорные расширения уравнений Лакса

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.