  Recently, Lax operator algebras appeared as a new class of higher genus
current type algebras. Based on I.Krichever's theory of Lax operators on
algebraic curves they were introduced by I. Krichever and O. Sheinman. These
algebras are almost-graded Lie algebras of currents on Riemann surfaces with
marked points (in-points, out-points, and Tyurin points). In a previous joint
article of the author with Sheinman the local cocycles and associated
almost-graded central extensions are classified in the case of one in-point and
one out-point. It was shown that the almost-graded extension is essentially
unique. In this article the general case of Lax operator algebras corresponding
to several in- and out-points is considered. In a first step it is shown that
they are almost-graded. The grading is given by the splitting of the marked
points which are non-Tyurin points into in- and out-points. Next,
classification results both for local and bounded cocycles are shown. The
uniqueness theorem for almost-graded central extensions follows. For this
generalization additional techniques are needed which are presented in this
article.
