Аннотация:
I would like to report on a cycle of works which came into existence due to an illuminating discussion with E. B. Vinberg in 2013.
Lax operator algebras emerged in the joint work by Krichever and the author (2007) where it was observed that the space of Lax operators with the spectral parameter on a Riemann surface invented by Krichever in 2001, possesses a structure of a Lie algebra. However, Lax operator algebras where constructed only for classical Lie algebras, and for several years it was not clear how to define them in terms of root systems. It was a challenging problem because of obvious relation of Lax operator algebras to fundamental questions of the theory of integrable systems and holomorphic vector bundles on Riemann surfaces. In particular, the last relation is based on the Tyurin parameters of holomorphic vector bundles, which in turn go back to matrix divisors invented by A.Weil in the work (1938) considered now as a starting point of the theory of holomorphic vector bundles. For the holomorphic G-bundles, where G is an arbitrary complex reductive group, matrix divisors are responsible for the "algebraic group part" of the theory.
In course of the above mentioned discussion E.B.Vinberg associated local conditions defining Lax operator algebras with Z-gradings of reductive Lie algebras. This has led to understanding that the three structures listed in the title are given by the same kind of data consisting of a Riemann surface with marked points, a complex reductive Lie algebra, a tuple of its Z-gradings associated with some of the marked points, and, in the case of integrable systems, of a positive divisor supported at the remainder of the points. The theory of finite-dimensional integrable systems (like Hitchin systems, gyroscopes, etc.) including their hierarchies and Hamiltonian theory, as well as the theory of matrix divisors of holomorphic G-bundles, are constructed in this general set-up now.
In the talk, I am going to define the main objects in terms of the above mentioned data, and formulate main relations between them.