Lax operator algebras and integrable systems

Lax operator algebras are introduced in [1] in connection with the notion of Lax operator with spectral parameter on a Riemann surface (earlier introduced by I. M. Krichever). These are algebras of currents defined on Riemann surfaces and taking values in the semi-simple or reductive Lie algebras. They are closely related to integrable systems like Hitchin systems, Calogero–Moser systems, classical gyroscopes, problems of flow around a solid body. In many respects, the Lax operator algebras are analogous to the Kac–Moody algebras. Non-twisted Kac–Moody algebras are Lax operator algebras on Riemann sphere with marked points 0, and $\infty$.
Up to the end of 2013 Lax operator algebras have been defined and constructed only for classical Lie algebras over $\mathbb C [1,2]$, and for the exceptional Lie algebra $G_2$, in terms of their matrix representations. A natural, and long standing question of their general construction in terms of root systems has been resolved in the beginning of this year [3]. It is a pleasant duty of the author to stress the role of E. B. Vinberg in the discussion of the question.
In the talk, we are going to give a general definition of Lax operator algebras in terms of gradings of semi-simple Lie algebras, formulate their basic properties. It will be stated a connection with Tyurin parameters of holomorphic vector bundles on Riemann surfaces. We are planning to formulate a general approach to construction of finite-dimensional Riemann surfaces based on the same circle of ideas.

References

1. I. M. Krichever, O. K. Sheinman, “Lax operator algebras”, Funct. Anal. Appl., 41:4 (2007), 284–294              
2. O. K. Sheinman, Current algebras on Riemann surfaces, De Gruyter Expositions in Mathematics, 58, Walter de Gruyter GmbH & Co, Berlin–Boston, 2012, 150 pp.  
3. O. K. Sheinman, “Lax operator algebras and gradings on semi-simple Lie algebras”, Transformation groups, arXiv: 1406.5017 (accepted)