Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2
Saïd Benayadi, Sofiane Bouarroudj, Quentin Ehret
TL;DR
The paper addresses the structure theory of Lie superalgebras in characteristic 2 by developing left-symmetric and left-alternative frameworks and introducing a cohomological method for Lagrangian extensions. It constructs a char-2 Lagrangian extension theory via flat torsion-free connections, proving that strongly polarized quasi-Frobenius Lie superalgebras arise as $T^*$- or $\Pi T^*$-extensions and is classified by a dedicated Lagrangian cohomology $XH^2_L$. The authors classify left-symmetric superalgebras in dimension 2, provide explicit 4-dimensional Lagrangian extensions, and supply extensive appendices on post-Lie structures and invariant data. Together, these results extend Bordmann–Baues–Cortés–Maeda frameworks to characteristic 2, yielding concrete classifications and cohomological tools with potential implications for modular Lie superalgebras and related geometric structures.
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.