Oleg Sheinman

Lax operator algebras and gradings on semi-simple Lie algebras

Lax operator algebras appeared as a unifying tool for construction, investigation of the Hamiltonian structure, and quantization of finite-dimensional integrable systems, like Hitchin systems, gyroscopes, and integrable cases of hydrodynamics of a solid body in a 2-dimensional flow. So far Lax operator algebras were known for classical simple and reductive Lie algebras, and for the exceptional Lie algebra G_2. The talk is devoted to a new general approach to Lax operator algebras, and their central extensions, given in terms of gradings of the corresponding semi-simple algebras, i.e. mainly in terms of their root systems. After joint work with E.B.Vinberg.