Cohomology of $\mathfrak{s}\mathfrak{l}(2)$ and its applications in prime characteristic

Over a field of characteristic $p>2,$ the first cohomology of the 3-dimensional simple Lie algebra $\mathfrak{s}\mathfrak{l}(2)$ 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 $\mathfrak{s}\mathfrak{l}(2)$ on any simple module are also be characterized.