Central extensions of Lax operator algebras

TL;DR

This paper classifies local cocycles and central extensions of Lax operator algebras, revealing that for simple finite-dimensional Lie algebras, the two-cohomology space is one-dimensional, with vector fields acting via covariant derivatives.

Contribution

It provides a classification of local cocycles and central extensions for Lax operator algebras, extending the understanding of their cohomological structure.

Findings

Two-cohomology space is one-dimensional for simple finite-dimensional Lie algebras.

Local cocycles and almost-graded central extensions are classified.

Vector fields act via covariant derivatives on Lax operator algebras.

Abstract

Lax operator algebras were introduced by Krichever and Sheinman as a further development of I.Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this article local cocycles and associated almost-graded central extensions are classified. It is shown that in the case that the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.