Restricted cohomology of restricted Lie superalgebras

TL;DR

This paper explores the restricted cohomology of restricted Lie superalgebras over algebraically closed fields of characteristic p>2, providing explicit descriptions and algebraic interpretations of low-dimensional cohomology spaces.

Contribution

It introduces a family of restricted model filiform Lie superalgebras and explicitly describes their low-dimensional cohomology spaces, revealing differences between ordinary and restricted cohomology.

Findings

Explicit description of 1-dimensional cohomology superspaces

Explicit description of 2-dimensional cohomology superspaces

Demonstration of equality and inequality between ordinary and restricted cohomology

Abstract

Suppose the ground field $\mathbb{F}$ is an algebraically closed field characteristic of $p>2$. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra $L_{p,p}^{\lambda}$ structures parameterized by elements $\lambda\in \mathbb{F}^{p}.$ We explicitly describe both the $1$-dimensional ordinary and restricted cohomology superspaces of $L_{p,p}^{\lambda}$ with coefficients in the $1$-dimensional trivial module and show that these superspaces are equal. We also describe the $2$-dimensional ordinary and restricted cohomology superspaces of $L_{p,p}^{\lambda}$ with coefficients in the $1$-dimensional trivial module and show that these superspaces are unequal.