Polyvector fields and polydifferential operators associated with Lie pairs
Autor: | Mathieu Stiénon, Ruggero Bandiera, Ping Xu |
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Rok vydání: | 2021 |
Předmět: |
lie algebroids
Lie algebroid Statistics::Theory Gerstenhaber algebra 01 natural sciences Combinatorics Mathematics::Probability Mathematics::K-Theory and Homology Mathematics::Quantum Algebra Mathematics - Quantum Algebra 0103 physical sciences FOS: Mathematics Quantum Algebra (math.QA) Gerstenhaber algebras homotopy lie algebras 0101 mathematics Mathematics::Representation Theory Mathematical Physics Mathematics Algebra and Number Theory Homotopy 010102 general mathematics Manifold Foliation Transfer (group theory) 010307 mathematical physics Geometry and Topology Isomorphism |
Zdroj: | Journal of Noncommutative Geometry. 15:643-711 |
ISSN: | 1661-6952 |
DOI: | 10.4171/jncg/416 |
Popis: | 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). Comment: [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 |
Databáze: | OpenAIRE |
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