The Second Cohomology Group of Elementary Quadratic Lie Superalgebras and Classifying a Subclass of 8-dimensional Solvable Quadratic Lie Superalgebras

Abstract

By definition, a quadratic Lie superalgebra is a Lie superalgebra endowed with a non-degenerate supersymmetric bilinear form which satisfies the even and invariant properties. In this paper we calculate all of the second cohomology group of elementary quadratic Lie superalgebras which have been classified in \cite{DU14} by applying the super-Poisson bracket on the super exterior algebra. Besides, we give the classification of 8-dimensional solvable quadratic Lie superalgebras having 6-dimensional indecomposable even part. The method is based on the double extension and classification results of adjoint orbits of the Lie algebra $\mathfrak{s}\mathfrak{p}(2)$.