Flat pseudo-Euclidean Leibniz superalgebras
Saïd Benayadi, Sofiane Bouarroudj, Hamza El Ouali
TL;DR
The paper develops a framework for flat pseudo-Euclidean Leibniz superalgebras by introducing pre-Leibniz superalgebras and a Levi-Civita product that connects the non-associative structure with a symmetric bilinear form. It proves that flatness is equivalent to a pre-left Leibniz structure and provides a detailed characterization, including parallel results for even and odd quadratic Leibniz superalgebras where flatness corresponds to 2-step nilpotent symmetric Leibniz algebras. The work extends T^*-type and Pi(T^*)-extensions to both even and odd quadratic settings and shows how these constructions classify reduced 2-step nilpotent cases. A central contribution is the double-extension program: every flat non-Lie left Leibniz superalgebra can be obtained by successive central extensions and semi-direct products from a flat Lie superalgebra, reducing the classification problem to the Lie case and offering explicit extension data and compatibility conditions. Overall, the results unify and extend Milnor-type ideas to the Leibniz superalgebra setting and provide a robust toolkit for constructing and classifying flat pseudo-Euclidean Leibniz superalgebras via iterative extensions.
Abstract
In this paper, we introduce pre-Lie and pre-Leibniz superalgebras, which generalize pre-Lie and pre-Leibniz algebras to the super setting. Additionally, we define a Levi-Civita product associated with a symmetric non-degenerate bilinear form on a non-associative superalgebra. This leads to the definition of flat pseudo-Euclidean left Leibniz superalgebras as those whose Levi-Civita product induces a pre-Leibniz structure. We study the structure of flat pseudo-Euclidean left Leibniz superalgebras and provide a characterization theorem. In the second part, we focus on quadratic Leibniz superalgebras and show that such a superalgebra is flat if and only if it is symmetric Leibniz and 2-step nilpotent. We further study the structure of quadratic 2-step nilpotent symmetric Leibniz superalgebras. Finally, we introduce the notion of double extension for flat pseudo-Euclidean (resp. Lie) left Leibniz superalgebras and prove that any flat pseudo-Euclidean non-Lie left Leibniz superalgebra can be obtained by a sequence of double extensions starting from a flat pseudo-Euclidean Lie superalgebra.