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?
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.
