Almost commutative $Q$-algebras and derived brackets

Abstract

We introduce the notion of almost commutative $Q$-algebras and demonstrate how the derived bracket formalism of Kosmann–Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaıntrob’s $Q$-manifold understanding of classical Lie algebroids. We show that the basic tenets of the theory of Lie algebroids carry over verbatim to the almost commutative world.