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, ${\mathfrak{h}}_{2m,n}$ and ${\mathfrak{ba}}_n$ with $m$ a nonnegative integer and $n$ a positive integer. The space of a "classical" Heisenberg Lie superalgebra ${\mathfrak{h}}_{2m,n}$ 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, ${\mathfrak{ba}}_n$, 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 ${\mathfrak{h}}_{2m,n}$ and ${\mathfrak{ba}}_n$ 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 ${\mathfrak{ba}}_n$ and we also make an attempt to determine the cohomology of ${\mathfrak{h}}_{2m,n}$ by computing it in a low-dimensional case.

19 pages

References in corpus (2)

• Superspace de Rham Complex and Relative Cohomology
• Lie superalgebra structures in cohomology spaces of Lie algebras with coefficients in the adjoint representation

Cited by in corpus (6)

• On cohomology of filiform Lie superalgebras
• On multipliers of pairs of Lie superalgebras
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• The Second Cohomology Group of Elementary Quadratic Lie Superalgebras and Classifying a Subclass of 8-dimensional Solvable Quadratic Lie Superalgebras
• On ${\mathrm{ID}}^{\ast }$-superderivations of Lie superalgebras
• Adjoint cohomology of two-step nilpotent Lie superalgebras