  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.
