  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
