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 𝕏 of a strong homotopy Lie–Rinehart algebra. It is natural to interpret 𝕏 as the (derived) Lie–Rinehart algebra of vector fields on the space P of integral manifolds of C. In this paper, we show that 𝕏 is embedded in an A∞-algebra 𝔻 of (normal) differential operators. It is natural to interpret 𝔻 as the (derived) associative algebra of differential operators on P. Finally, we speculate about the interpretation of 𝔻 as the universal enveloping strong homotopy algebra of 𝕏.