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

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.

References in corpus (3)

• Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix
• Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras
• Left-symmetric Superalgebras on Special Linear Lie Superalgebras