Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

Abstract

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given pde . The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the pde s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer–Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.