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

TL;DR

This paper introduces left-symmetric and left alternative structures on superspaces in characteristic 2, classifies them in dimension 2, and develops a cohomological framework for Lagrangian extensions of Lie superalgebras, with explicit examples in dimension 4.

Contribution

It defines and classifies new algebraic structures in characteristic 2 and presents a cohomological method for Lagrangian extensions of Lie superalgebras.

Findings

Classification of left-symmetric structures in dimension 2

Lagrangian extensions correspond to a specific cohomology space

Explicit descriptions of all dimension 4 Lagrangian extensions

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.