Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2

Saïd Benayadi; Sofiane Bouarroudj; Quentin Ehret

doi:10.1016/j.jpaa.2025.108086

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arXiv · 2501.15432

Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2

Saïd Benayadi·Sofiane Bouarroudj·Quentin Ehret

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Abstract

The purpose of this paper is twofold. First, we introduce the notions of left-symmetric and left alternative structures on superspaces in characteristic 2. We describe their main properties and classify them in dimension 2. We show that left-symmetric structures can be queerified if and only if they are left-alternative. Secondly, we present a method of Lagrangian extension of Lie superalgebras in characteristic 2 with a flat torsion-free connection. We show that any strongly polarized quasi-Frobenius Lie superalgebra can be obtained as a Lagrangian extension. Further, we demonstrate that Lagrangian extensions are classified by a certain cohomology space that we introduce. To illustrate our constructions, all Lagrangian extensions in dimension 4 have been described.

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Saïd Benayadi, Sofiane Bouarroudj, Quentin Ehret. 2025-01-26. Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2. https://doi.org/10.1016/j.jpaa.2025.108086

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