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

## TL;DR

This paper explores how exotic cohomology of infinite-dimensional Lie algebras can be used to identify Lax representations for integrable PDEs, providing a new internal criterion for integrability.

## Contribution

It demonstrates that Maurer-Cartan forms of extended Lie algebras with nontrivial exotic cohomology generate Lax representations, linking algebraic cohomology to integrability conditions.

## Key findings

- Maurer-Cartan forms produce Lax representations for known integrable systems
- New integrable PDEs are identified through exotic cohomology analysis
- Exotic cohomology offers a criterion for integrability based on algebraic properties

## 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.