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 $\mathbb{X}$.

28 pages, comments welcome

References in corpus (5)

• On the category of Lie n-algebroids
• Lie-Rinehart algebras, descent, and quantization
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• A universal enveloping for L-infinity algebras
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Cited by in corpus (12)

• Representations of Homotopy Lie-Rinehart Algebras
• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• Shifted derived Poisson manifolds associated with Lie pairs
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Hopf algebras arising from dg manifolds
• Polyvector fields and polydifferential operators associated with Lie pairs
• Keller admissible triples and Duflo theorem
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• $A_{\infty }$-Algebras from Lie Pairs
• Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair