Double and Lagrangian extensions for quasi-Frobenius Lie superalgebras

TL;DR

This paper explores the structure and classification of quasi-Frobenius Lie superalgebras, introducing double and Lagrangian extensions, and classifying all 4-dimensional cases as solvable.

Contribution

It introduces the concepts of double and Lagrangian extensions for quasi-Frobenius Lie superalgebras and classifies all 4-dimensional cases, showing they are solvable.

Findings

Every 4-dimensional quasi-Frobenius Lie superalgebra is solvable.

All such superalgebras can be constructed via double extensions.

Lagrangian extensions are classified by a new cohomology space.

Abstract

A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of $T^*$-extensions (or Lagrangian extensions) of Lie superalgebras, and show that they are classified by a certain cohomology space we introduce. Several examples are provided to illustrate our construction.