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Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no experience with this, so not sure if this is a too easy question.

One can replace p-group algebra by local selfinjective finite dimensional algebra if it is too easy. Can the simple lie algebras realised as first hochschild cohomology of such (or other) algebras?

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I'm not an expert but it looks like some interesting simple Lie algebras do appear this way, yet one does not know how many. If the base field $k$ is $\mathbb{F}_p$, where $p$ is odd, and $G=\mathbb{Z}_p$ is a cyclic group of order $p$, then $HH^*(kG,kG)$ is recently described along with all relevant operations (even as a BV-algebra). It seems that the Lie algebra $HH^1(\mathbb{Z}_p,\mathbb{Z}_p)$ is isomorphic to the Witt algebra $W(1,\underline{1})$ which is the derivation algebra of the truncated polynomial ring $k[X]/(X^p)$. The Witt algebra belongs to a large family of finite dimensional simple Lie algebras called Lie algebras of Cartan type (there are four types: $W, S, H$ and $K$). Markus Linckelmann showed recently that some Cartan type algebras of type $W$ can be realised as $HH^1(B,B)$ where $B$ is a block of a finite group algebra. In the more general setting of a self-injective finite dimensional local algebra, the answer is unknown.

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