Flat quasi-Frobenius Lie superalgebras

TL;DR

This paper studies flat quasi-Frobenius Lie superalgebras, introduces the concept of flat double extensions, and classifies such superalgebras of small dimension, revealing their nilpotent and degenerate center structure.

Contribution

It introduces the notion of flat double extensions for classifying flat quasi-Frobenius Lie superalgebras and provides a complete classification for low-dimensional cases.

Findings

Flat quasi-Frobenius Lie superalgebras are nilpotent.

Such superalgebras have degenerate centers.

They can be constructed via flat double extensions.

Abstract

A non-associative superalgebra is called pre-symplectic if it is equipped with a non-degenerate, anti-symmetric bilinear form. It is called quasi-Frobenius if, in addition, is a Lie superalgebra and the form is closed. We introduce the Levi-Civita product associated with pre-symplectic superalgebras and establish its existence and uniqueness. We then introduce the symplectic product associated with quasi-Frobenius Lie superalgebras. We prove that while such a product always exists, it is not unique. We therefore define a natural symplectic product that depends only on the Lie structure and the bilinear form. When the curvature of this product vanishes, the superalgebra is called a flat quasi-Frobenius Lie superalgebra. In this paper, we study flat quasi-Frobenius Lie superalgebras and introduce the notion of a flat double extension. We prove that the double extension process characterizes such superalgebras. More precisely, every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be obtained by a sequence of flat double extensions starting from an abelian one (resp. the trivial one). Moreover, we show that every flat quasi-Frobenius Lie superalgebra is nilpotent with a degenerate center. We apply our results to obtain a complete classification of such superalgebras of total dimension at most five.