Flat quasi-Frobenius Lie superalgebras
Sofiane Bouarroudj, Hamza El Ouali
TL;DR
The paper develops a comprehensive framework for flat quasi-Frobenius Lie superalgebras by linking Levi-Civita and natural symplectic products to left-symmetric structures. It introduces flat double extensions as a core construction tool, proving that every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be built from an abelian (resp. trivial) base via successive flat extensions, and it establishes that such algebras are necessarily nilpotent with degenerate centers in the non-abelian case. A full classification is achieved up to total dimension five, with explicit descriptions of the four-dimensional non-abelian models $ oldsymbol{g}^1,oldsymbol{g}^2,oldsymbol{g}^3,oldsymbol{g}^4$ and two-parameter families in five dimensions, all realized as flat double extensions. The results unify orthosymplectic and periplectic settings, providing structural insight into solvability, nilpotency, and the role of flatness in the associated left-symmetric structures. This advances understanding of flat affine and symplectic-supergeometry avenues and yields a complete low-dimensional landscape useful for further representation-theoretic and geometric investigations.
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.