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

Almost-graded central extensions of Lax operator algebras