Narrow Results

Suppose that $L$ is a Lie superalgebra over a field $\mathbb{F}$ 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.