On the Lie algebra structure of integrable derivations

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra A$A$ forms a Lie algebra, and a restricted Lie algebra if A$A$ contains a field of characteristic p$p$. We deduce that the space of integrable classes in HH1(A)${\operatorname{HH}}<^>1(A)$ forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self-injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group.