The Multiplier and Cohomology of Lie Superalgebras

Abstract

Suppose that [Formula: see text] is a Lie superalgebra over a field [Formula: see text] of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of [Formula: see text] is constructed and is proved to be exact. Moreover, the multiplier of [Formula: see text] is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of [Formula: see text]. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when [Formula: see text] is nilpotent and finite-dimensional.