Lax operator algebras and gradings on semi-simple Lie algebras

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 $\mathcal C$ [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], and is the main subject of the present talk.

References

1. 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 (to appear) , arXiv: 1406.5017