Polyvector fields and polydifferential operators associated with Lie pairs

Abstract

We prove that the spaces $\mathrm{tot}(\Gamma ({\Lambda }^{\bullet }{A}^{\vee }){\otimes }_R{\mathcal{T}}_{poly}^{\bullet })$ and $\mathrm{tot}(\Gamma ({\Lambda }^{\bullet }{A}^{\vee }){\otimes }_R{\mathcal{D}}_{poly}^{\bullet })$ 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 ${\mathbb{H}}_{\mathrm{CE}}^{\bullet }(A,{\mathcal{T}}_{poly}^{\bullet })$ and ${\mathbb{H}}_{\mathrm{CE}}^{\bullet }(A,{\mathcal{D}}_{poly}^{\bullet })$ 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).