# Polyvector fields and polydifferential operators associated with Lie pairs

## TL;DR

This paper constructs canonical $L_$ algebra structures on spaces related to Lie pairs, enabling the definition of Gerstenhaber algebra structures on their cohomologies, using homotopy transfer and Fedosov dg Lie algebroids.

## Contribution

It introduces a method to endow spaces associated with Lie pairs with $L_$ algebra structures, providing new tools for their deformation theory and cohomological analysis.

## Key findings

- Spaces carry canonical $L_$ algebra structures
- Unique Gerstenhaber algebra structures on cohomology groups
- Application of homotopy transfer and Fedosov dg Lie algebroids

## Abstract

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