Derivations from the even parts into the odd parts for Lie superalgebras W and S

TL;DR

This paper investigates derivations from the even parts of certain Lie superalgebras into their odd parts, providing explicit descriptions of these derivation spaces over fields with characteristic greater than three.

Contribution

It determines the derivation spaces from the even parts of Witt and S superalgebras into their odd parts using reduction on -gradations, a novel approach in this context.

Findings

Explicit description of er(W, W_{ar{1}}) and er(S, W_{ar{1}})

Determination of er(S, S_{ar{1}})

Application of -gradation reduction method

Abstract

Let $\mathcal{W}$ and $\mathcal{S}$ denote the even parts of the general Witt superalgebra $W$ and the special superalgebra $S$ over a field of characteristic $ p>3,$ respectively. In this note, using the method of reduction on $\mathbb{Z}$-gradations, we determine the derivation space $\mathrm{Der}(\mathcal{W}, W_{\bar{1}})$ from $\mathcal{W}$ into $W_{\bar{1}} $ and the derivation space $\mathrm{Der}(\mathcal{S}, W_{\bar{1}})$ from $\mathcal{S}$ into $W_{\bar{1}}. $ In particular, the derivation space $\mathrm{Der}(\mathcal{S}, S_{\bar{1}})$ is determined.