The multiplier and cohomology of Lie superalgebras

TL;DR

This paper develops a cohomology framework for Lie superalgebras, establishing exact sequences and linking multipliers to second cohomology groups, thus advancing the algebraic understanding of their extensions.

Contribution

It introduces 5-sequences of cohomology for central extensions of Lie superalgebras and proves their exactness, connecting multipliers to second cohomology groups.

Findings

Constructed 5-sequences of cohomology for Lie superalgebras

Proved the exactness of these cohomology sequences

Showed multipliers are isomorphic to second cohomology groups

Abstract

In this paper, all (super)algebras are over a field $\mathbb{F}$ of characteristic different from $2, 3$. We construct the so-called 5-sequences of cohomology for central extensions of a Lie superalgebra and prove that they are exact. Then we prove that the multipliers of a Lie superalgebra are isomorphic to the second cohomology group with coefficients in the trivial module for the Lie superalgebra under consideration.