On the Lie algebra structure of integrable derivations

TL;DR

This paper demonstrates that integrable derivations on Artin algebras form a Lie algebra, which remains invariant under certain equivalences, and explores related properties and limitations of integrable derivations.

Contribution

It establishes the Lie algebra structure of integrable derivations on Artin algebras and their invariance under derived and stable Morita equivalences.

Findings

Integrable derivations form a Lie algebra on Artin algebras.

The Lie algebra structure is preserved under derived and stable equivalences.

Negative results on existing questions about integrable derivations.

Abstract

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra $A$ forms a Lie algebra, and a restricted Lie algebra if $A$ contains a field of characteristic $p$. We deduce that the space of integrable classes in $\HH^1(A)$ 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.