Central extensions of Lax operator algebras

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.