The Multiplier and Cohomology of Lie Superalgebras

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