Homogeneous quadratic Lie super algebras

TL;DR

This paper introduces a generalized double extension method for homogeneous quadratic Lie super algebras, encompassing even and odd cases, and demonstrates its ability to construct all indecomposable, non-simple instances, including previously unaddressed cases.

Contribution

It extends the double extension framework to homogeneous quadratic Lie super algebras, unifying and broadening previous approaches.

Findings

Any indecomposable, non-simple homogeneous quadratic Lie super algebra can be constructed via this double extension.

The new method recovers all previously studied cases and introduces new ones not accessible by earlier methods.

The approach unifies even and odd cases within a single framework.

Abstract

In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by means of this type of double extension. We also show that with this construction we can recover previously studied cases as well as some other which can not be recover with former versions of double extensions.