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

中文翻译：

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与李对相关的多向量场和多微分算子

我们证明空间 $\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{T}_{poly}^{\bullet}}}\ big)$ 和 $\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{D}_{poly}^{\bullet}}}\big )$ 与一个李对 $(L,A)$ 相关联，每个都携带一个 $L_\infty$ 代数结构规范到 $L_\infty$ 同构，以恒等映射作为线性部分。这两个空间分别用作 Lie 对 $(L,A)$ 上形式多向量域和形式多微分算子的空间的替代。因此，$\mathbb{H}^{\bullet}_{\operatorname{CE}}(A,{\mathcal{T}_{poly}^{\bullet}})$ 和 $\mathbb{H} ^{\bullet}_{\operatorname{CE}}(A,{\mathcal{D}_{poly}^{\bullet}})$ 承认独特的 Gerstenhaber 代数结构。