Lax operator algebras and gradings on semi-simple Lie algebras

In 2001 I. M. Krichever had introduced the Lax operators with spectral parameter on a Riemann surface, in terms of the Tyurin parameters of holomorphic bundles, and applied this construction to the investigation of systems of Hitchin/Calogero type and their generalizations, in particular for proving that they are Hamiltonian. In 2006, in the joint work of I. M. Krichever and the author the multiplicative properties of the Lax operators of this class have been found out, and their analogs taking values in the classical Lie algebras have been constructed. Considered as meromorphic functions of the spectral parameter these operators form infinite-dimensional Lie algebras directly generalizing loop algebras. They are almost-graded, and possess non-trivial central extensions. Later, the Lax operator algebra for the exceptional simple Lie algebra $G_2$ had been constructed. It became clear that closeness of Lax operators with respect to the commutator, and the Hamiltonian property of the corresponding equations are equivalent to the same relations on the Tyurin parameters. Some properties of the Lax operators, such as difference in the orders of their poles in the Tyurin points, had no explanation. In the talk, a general construction of Lax operator algebras for an arbitrary complex semi-simple Lie algebra will be considered, enabling us to give general proofs of their basic properties. It will be shown how to retrieve the Tyurin parameters in frame of this construction which probably means that there is a connection between holomorphic bundles and semi-simple Lie algebras.