The First Cohomology of \(\mathfrak{s}\mathfrak{l}(2)\) and Its Applications in Prime Characteristic

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