Lax operator algebras and integrable systems

Abstract: A new class of infinite-dimensional Lie algebras given a name of Lax operator algebras, and the related unifying approach to finite-dimensional integrable systems with spectral parameter on a Riemann surface, such as Calogero--Moser and Hitchin systems, are presented. In particular, our approach includes (the non-twisted) Kac--Moody algebras and integrable systems with rational spectral parameter. The presentation is based on the quite simple ideas related to gradings of semisimple Lie algebras, and their interaction with the Riemann--Roch theorem. The basic properties of the Lax operator algebras, as well as the basic facts of the theory of integrable systems in question, are treated (and proven) from this general point of view, in particular existence of commutative hierarchies and their Hamiltonian properties. We conclude with an application of the Lax operator algebras to prequantization of finite-dimensional integrable systems.