Lagrangian extensions and left symmetric structures on the four-dimensional real Lie superalgebras

TL;DR

This paper classifies four-dimensional real Lie superalgebras that can be constructed as Lagrangian extensions and explores their left-symmetric and Novikov structures, revealing new algebraic properties.

Contribution

It identifies which four-dimensional real Lie superalgebras are Lagrangian extensions and analyzes their left-symmetric and Novikov structures, providing a comprehensive classification.

Findings

Most of the classified superalgebras admit left-symmetric structures.

All but two of these superalgebras are Novikov superalgebras.

The paper offers a detailed characterization of these algebraic structures.

Abstract

Over real numbers, Backhouse classified all four-dimensional Lie superalgebras. From this list, we will investigate those Lie superalgebras that can be obtained as Lagrangian extensions. Moreover, we investigate left-symmetric structures on these Lie superalgebras. Furthermore, except for two of them, they are all Novikov superalgebras.