Lax operator algebras and gradings on semi-simple Lie algebras

TL;DR

This paper constructs Lax operator algebras for semi-simple Lie algebras with gradings over Riemann surfaces, classifies their central extensions, and relates this to previous Tyurin parameter methods.

Contribution

It introduces a new construction of Lax operator algebras for graded semi-simple Lie algebras on Riemann surfaces and classifies their central extensions.

Findings

Constructed Lax operator algebras for arbitrary semi-simple Lie algebras with gradings.

Classified the central extensions of these Lax operator algebras.

Established a relation to the Tyurin parameters approach.

Abstract

A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded structures, and classification of the central extensions of Lax operator algebras are given. A relation to the earlier approach based on the Tyurin parameters is established.