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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, h 2m,n and ba n with m a nonnegative integer and n a positive integer. The space of a "classical" Heisenberg Lie superalgebra h 2m,n is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a onedimensional space of values of this form constituting the even center. The other super analog of the Heisenberg Lie algebra, 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 h 2m,n and 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 ba n and we also make an attempt to determine the cohomology of h 2m,n by computing it in a lowdimensional case.
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, h 2m,n and ba n with m a nonnegative integer and n a positive integer. The space of a "classical" Heisenberg Lie superalgebra h 2m,n is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a onedimensional space of values of this form constituting the even center. The other super analog of the Heisenberg Lie algebra, 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 h 2m,n and 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 ba n and we also make an attempt to determine the cohomology of h 2m,n by computing it in a lowdimensional case.
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