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, 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.

The Matrix Representation of Quasi Centroids of Heisenberg Superalgebra

“…They are divided into two types -even and odd Heisenberg Lie superalgebras. [6] Definition 2.1. ( [1]) Definition of Lie superalgebra: Let L = L0 ⊕ L1 be a Z2− order linear space, if any Z2− homogeneous element satisfies super antisymmetry,…”

On multipliers of pairs of Lie superalgebras

“…Recall from [2,3] that a finite dimensional Lie (super)algebra L is called Heisenberg, if L 2 = Z(L) and dim L 2 = 1. A Heisenberg Lie algebra, denoted by H(m), has dimension 2m + 1 with a basis {x 1 , .…”

“…Kac (Kats) [4] in 1970. Developing the properties of Lie superalgebras has been of some interest in mathematics and theoretical physics for the last 40 years (see [2,9,10,15] for more information).…”

On cohomology of filiform Lie superalgebras

“…In this section, we introduce the definitions of cohomology and divided power cohomology of g with coefficients in the trivial module. For more details, the reader is referred to [13,15,16]. Let g be a finite-dimensional Lie superalgebra, denote by g * the dual superspace of g. Fix an ordered basis of g…”

“…In this section, we introduce the definitions of cohomology and divided power cohomology of g with coefficients in the trivial module. For more details, the reader is referred to [13,15,16].…”