Cohomology of $\frak{sl}(2)$ and its applications in prime characteristic

TL;DR

This paper computes the first cohomology of the simple Lie algebra sl(2) over fields of prime characteristic, revealing that Whitehead's first lemma does not hold in this setting, and characterizes derivations on simple modules.

Contribution

It determines the first cohomology of sl(2) in prime characteristic and shows the failure of Whitehead's first lemma, providing new insights into Lie algebra cohomology in this context.

Findings

Whitehead's first lemma is not valid in prime characteristic

First cohomology of sl(2) with simple modules is explicitly determined

Derivations and 2-local derivations of sl(2) are characterized

Abstract

Over a field of characteristic $p>2,$ the first cohomology of the 3-dimensional simple Lie algebra $\frak{sl}(2)$ with coefficients in all simple modules is determined, which implies Whitehead's first lemma is not true in prime characteristic. As applications, derivations and 2-local derivations of $\frak{sl}(2)$ on any simple module are also be characterized.