Title: 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
use of the exotic cohomology of the symmetry algebras opens a way to formulate
such conditions in internal terms of the PDEs 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.