Lie superalgebras of Krichever–Novikov type and their central extensions

Abstract

Classically important examples of Lie superalgebras have been constructed starting from the Witt and Virasoro algebra. In this article we consider Lie superalgebras of Krichever–Novikov type. These algebras are multi-point and higher genus equivalents. The grading in the classical case is replaced by an almost-grading. The almost-grading is determined by a splitting of the set of points were poles are allowed into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to an almost-grading, it is shown that there is up to rescaling and equivalence a unique non-trivial central extension. It is given explicitly. Furthermore, a complete classification of bounded cocycles (with respect to the almost-grading) is given.