Over an algebraically closed field of characteristic p > 2, the first cohomology of the 3-dimensional simple Lie algebra 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 sl(2) on any simple module are also characterized.
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Wang et al. (2024) studied this question.
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