Multipoint Lax operator algebras: almost-graded structure and central extensions

Abstract

Recently, Lax operator algebras appeared as a new class of higher genus current-type algebras. Introduced by Krichever and Sheinman, they were based on Krichever's theory of Lax operators on algebraic curves. 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 with Sheinman, the author classified the local cocycles and associated almost-graded central extensions 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. As a first step they are shown to be almost-graded. The grading is given by splitting the marked points which are non-Tyurin points into in- and out-points. Next, classification results both for local and bounded cocycles are proved. The uniqueness theorem for almost-graded central extensions follows. To obtain this generalization additional techniques are needed which are presented in this article. Bibliography: 30 titles.