Almost Commutative Q-algebras and Derived brackets

TL;DR

This paper introduces almost commutative Q-algebras, extending the derived bracket formalism and Lie algebroid theory to a new algebraic setting, bridging classical and noncommutative geometry.

Contribution

It generalizes the derived bracket formalism to almost commutative Q-algebras and constructs almost commutative Lie algebroids, preserving core Lie algebroid properties.

Findings

Derived bracket formalism extends to almost commutative Q-algebras

Construction of almost commutative Lie algebroids

Core Lie algebroid principles hold in the new setting

Abstract

We introduce the notion of \emph{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 Va\u{\i}ntrob'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.