Polyvector fields and polydifferential operators associated with Lie pairs

Abstract

We prove that the spaces $\mathrm{tot}(\Gamma ({\Lambda }^{\bullet }{A}^{\vee }{\otimes }_R{\mathcal{T}}_{\mathrm{poly}}^{\bullet })$ and $\mathrm{tot}(\Gamma ({\Lambda }^{\bullet }{A}^{\vee }){\otimes }_R{\mathcal{D}}_{\mathrm{poly}}^{\bullet })$ associated with a Lie pair $(L,A)$ each carry an $L_{\infty }$ algebra structure canonical up to an $L_{\infty }$ isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair $(L,A)$. Consequently, both ${\mathbb{H}}_{\mathrm{CE}}^{\bullet }(A,{\mathcal{T}}_{\mathrm{poly}}^{\bullet })$ and ${\mathbb{H}}_{\mathrm{CE}}^{\bullet }(A,{\mathcal{D}}_{\mathrm{poly}}^{\bullet })$ admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold).

[v2] 50 pages, paper was expanded; [v1] Paper arXiv:1605.09656v1 was expended and split into two papers. The first part is arXiv:1605.09656v2. The second part is the present paper. A new result addressing uniqueness of the constructed structures has been added

References in corpus (6)

• The Atiyah class of a dg-vector bundle
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Atiyah classes and dg-Lie algebroids for matched pairs
• Formality for g-manifolds
• L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras

Cited by in corpus (3)

• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Hochschild cohomology of dg manifolds associated to integrable distributions
• BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties