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

TL;DR

This paper computes the second cohomology groups of elementary quadratic Lie superalgebras and classifies certain 8-dimensional solvable quadratic Lie superalgebras using super-Poisson brackets and double extension methods.

Contribution

It provides the first calculation of second cohomology groups for classified elementary quadratic Lie superalgebras and classifies a specific subclass of 8-dimensional solvable quadratic Lie superalgebras.

Findings

All second cohomology groups of elementary quadratic Lie superalgebras are computed.

A classification of 8-dimensional solvable quadratic Lie superalgebras with 6-dimensional indecomposable even part is provided.

Methodology involves super-Poisson brackets and double extension techniques.

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 $\mathfrak{s}\mathfrak{p}(2)$.