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Recently, Lax operator algebras appeared as a new class of higher genuscurrent type algebras. Based on I.Krichever's theory of Lax operators onalgebraic curves they were introduced by I. Krichever and O. Sheinman. Thesealgebras are almost-graded Lie algebras of currents on Riemann surfaces withmarked points (in-points, out-points, and Tyurin points). In a previous jointarticle of the author with Sheinman the local cocycles and associatedalmost-graded central extensions are classified in the case of one in-point andone out-point. It was shown that the almost-graded extension is essentiallyunique. In this article the general case of Lax operator algebras correspondingto several in- and out-points is considered. In a first step it is shown thatthey are almost-graded. The grading is given by the splitting of the markedpoints which are non-Tyurin points into in- and out-points. Next,classification results both for local and bounded cocycles are shown. Theuniqueness theorem for almost-graded central extensions follows. For thisgeneralization additional techniques are needed which are presented in thisarticle.

Multipoint Lax operator algebras. Almost-graded structure and central extensions