Algebraic structures related to integrable differential equations

TL;DR

This survey explores algebraic structures underpinning integrable differential equations, focusing on Lax representations, loop algebras, and bi-Hamiltonian frameworks, revealing new insights into classical models like Calogero-Moser.

Contribution

It provides a comprehensive description of algebraic structures such as loop algebras and affine Dynkin diagrams related to integrable equations, including new connections to bi-Hamiltonian systems.

Findings

Lax representations expressed via vector space decompositions

Examples of complementary subalgebras linked to integrable models

Bi-Hamiltonian origin of elliptic Calogero-Moser models

Abstract

The survey is devoted to algebraic structures related to integrable ODEs and evolution PDEs. A description of Lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of Taylor series and a complementary subalgebra. Examples of complementary subalgebras and corresponding integrable models are presented. In the framework of the bi-Hamiltonian approach compatible associative algebras related affine Dynkin diagrams are considered. A bi-Hamiltonian origin of the classical elliptic Calogero-Moser models is revealed.