Almost-graded central extensions of Lax operator algebra

TL;DR

This paper studies the structure of Lax operator algebras, a class of infinite-dimensional Lie algebras from geometry, focusing on their almost-graded central extensions and classifying their unique cocycles.

Contribution

It provides a classification and explicit description of almost-graded central extensions of Lax operator algebras, establishing uniqueness for simple underlying Lie algebras.

Findings

Explicit forms of defining cocycles are given.

Up to equivalence, there is a unique non-trivial almost-graded central extension for simple Lie algebras.

Results extend the understanding of algebraic structures related to Riemann surfaces.

Abstract

Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for $\gl(n)$, with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. These results are joint work with Oleg Sheinman. This is an extended write-up of a talk presented at the 5 th Baltic-Nordic AGMP Workshop: Bedlewo, 12-16 October, 2009