Algebraic structures related to integrable differential equations

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.