On the Strong Homotopy Associative Algebra of a Foliation

Abstract

An involutive distribution $C$ on a smooth manifold $M$ is a Lie-algebroid acting on sections of the normal bundle $TM/C$. It is known that the Chevalley-Eilenberg complex associated to this representation of $C$ possesses the structure $\mathbb{X}$ of a strong homotopy Lie-Rinehart algebra. It is natural to interpret $\mathbb{X}$ as the (derived) Lie-Rinehart algebra of vector fields on the space $\mathbb{P}$ of integral manifolds of $C$. In this paper, I show that $\mathbb{X}$ is embedded in a strong homotopy associative algebra $\mathbb{D}$ of (normal) differential operators. It is natural to interpret $\mathbb{D}$ as the (derived) associative algebra of differential operators on $\mathbb{P}$. Finally, I speculate about the interpretation of $\mathbb{D}$ as the universal enveloping strong homotopy algebra of $\mathbb{X}$.