Lie superalgebras of Krichever-Novikov type and their central extensions

TL;DR

This paper explores Lie superalgebras of Krichever-Novikov type, extending classical structures to higher genus and multiple points, and classifies their central extensions and cocycles.

Contribution

It introduces a classification of central extensions for Krichever-Novikov type Lie superalgebras and explicitly constructs the unique non-trivial extension.

Findings

Unique non-trivial central extension up to rescaling and equivalence

Explicit construction of the central extension

Complete classification of bounded cocycles

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.