Maximal Subalgebras for Modular Graded Lie Superalgebras of Odd Cartan Type

The purpose of this paper is to determine all maximal graded subalgebras of the four infinite series of finite-dimensional graded Lie superalgebras of odd Cartan type over an algebraically closed field of characteristic $p>3$. All maximal graded subalgebras consist of three types (\MyRoman{1}), (\MyRoman{2}) and (\MyRoman{3}). Maximal graded subalgebras of type (\MyRoman{3}) fall into reducible maximal graded subalgebras and irreducible maximal graded subalgebras. In this paper we classify maximal graded subalgebras of types (\MyRoman{1}), (\MyRoman{2}) and reducible maximal g raded subalgebras.The classification of irreducible maximal graded subalgebras is reduced to that of the irreducible maximal subalgebras of the classical Lie superalgebra $\mathfrak{p}(n)$.Comment: For the final version, see Transformation Groups 20(4)(2015)1075--110

Cohomology of Heisenberg Lie Superalgebras

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.Comment: 19 page