Hom-Lie superalgebras in characteristic 2

TL;DR

This paper develops the structure, classification, and cohomology theory of Hom-Lie superalgebras in characteristic 2, introducing new notions of restrictedness and queerification, and establishing deformation theory.

Contribution

It introduces a new restrictedness concept, classifies low-dimensional cases, and develops a cohomology and deformation theory for Hom-Lie superalgebras in characteristic 2.

Findings

Classification of low-dimensional Hom-Lie superalgebras

Introduction of a new restrictedness notion

Development of cohomology and deformation theories

Abstract

The main goal of this paper is to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, $\alpha^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from one given by Guan and Chen. This definition is inspired by the process of queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we develop a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we establish a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology.