Cohomology of Heisenberg Lie Superalgebras

Abstract

Suppose the ground field to be algebraically closed and of characteristic different from $2$ and $3$. All Heisenberg Lie superalgebras consist of two super versions of the Heisenberg Lie algebras, $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with $m$ a nonnegative integer and $n$ a positive integer. The space of a "classical" Heisenberg Lie superalgebra $\frak{h}_{2m,n}$ is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a one-dimensional space of values of this form constituting the even center. The other super analog of the Heisenberg Lie algebra, $\frak{ba}_n$, is constructed by means of a non-degenerate anti-supersymmetric odd bilinear form with values in the one-dimensional odd center. In this paper, we study the cohomology of $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with coefficients in the trivial module by using the Hochschild-Serre spectral sequences relative to a suitable ideal. In characteristic zero case, for any Heisenberg Lie superalgebra, we determine completely the Betti numbers and associative superalgebra structure for their cohomology. In characteristic $p>3$ case, we determine the associative superalgebra structures for the divided power cohomology of $\frak{ba}_n$ and we also make an attempt to determine the cohomology of $\frak{h}_{2m,n}$ by computing it in a low-dimensional case.