Structure of Flat Quadratic Quasi-Frobenius Lie Superalgebras via Double Extensions

TL;DR

This paper classifies and constructs flat quadratic quasi-Frobenius Lie superalgebras using double extensions, introducing planar extensions for mixed parity cases and providing explicit low-dimensional examples.

Contribution

It introduces the concept of flat quadratic double and planar extensions for Lie superalgebras, expanding the methods to construct and classify these structures.

Findings

Any such Lie superalgebra can be built from the trivial algebra via double extensions.

Planar double extensions are introduced for mixed parity structures.

Complete classification of low-dimensional cases is provided.

Abstract

A flat quadratic quasi-Frobenius Lie superalgebra is a quadratic Lie superalgebra equipped with an additional symplectic structure that is flat with respect to the natural symplectic product. In this paper, we introduce the notion of a flat quadratic double extension of a flat quadratic quasi-Frobenius Lie superalgebra, in the cases where both the symplectic structure and the quadratic structure are either even or odd. We show that, over an algebraically closed field, any such Lie superalgebra can be constructed through a sequence of flat quadratic double extensions starting from the trivial algebra $\{0\}$. Moreover, when the quadratic and symplectic structures have different parity, we introduce the notion of a planar double extension, which constitutes the main novelty of this paper. In this case, we prove that such Lie superalgebras have total dimension $4n$. Finally, we classify flat quadratic quasi-Frobenius Lie superalgebras of dimension at most four and present explicit examples in dimensions six and eight.