Formality and Kontsevich–Duflo type theorems for Lie pairs

Liao, Hsuan-Yi; Stiénon, Mathieu; Xu, Ping

doi:10.1016/j.aim.2019.04.047

Cited by 14 publications

(14 citation statements)

References 68 publications

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“…As an immediate consequence, applying the Kontsevich-Duflo type theorem for the dg manifold (T 0,1 X [1], ∂) [23,38], we recover Kontsevich-Duflo theorem for complex manifolds, a theorem first proved by Kontsevich (for associative algebras only) in [20], Calaque and Van den Bergh in [5] and recovered by Liao, Stiénon and Xu using Lie pairs in [24].…”

Section: Furthermore We Provesupporting

confidence: 56%

“…Note that the notations (T p poly (F [1])) q and (D p poly (F [1])) q in this paper follow those in [11,38], which are shifted by degree (+1) comparing to [23]. Similarly, the notations T n poly (B) and D n poly (B) in this paper are the same but up to a degree shift as the ones in [24].…”

Section: Kontsevich-duflo Type Isomorphisms For Integrable Distributionsmentioning

confidence: 93%

“…This section is devoted to the proof of main theorems. We start by recalling some existing results from [14,23,24]. Note that the notations (T p poly (F [1])) q and (D p poly (F [1])) q in this paper follow those in [11,38], which are shifted by degree (+1) comparing to [23].…”

Section: Kontsevich-duflo Type Isomorphisms For Integrable Distributionsmentioning

confidence: 99%

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Hochschild cohomology of dg manifolds associated to integrable distributions

Chen,

Xiang,

2021

Preprint

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For the field K = R or C, and an integrable distribution F ⊆ T K M = TM ⊗ R K on a smooth manifold M , we study the Hochschild cohomology of the dg manifold (F [1], dF ) and establish a canonical isomorphism with the Hochschild cohomology of the transversal polydifferential operators of F . In particular, for the dg manifold (T 0,1 X [1], ∂) associated with a complex manifold X, we prove that it is canonically isomorphic to the Hochschild cohomology HH • (X) of the complex manifold. As an application, we show that the Kontsevich-Duflo type theorem for the dg manifold (T 0,1 X [1], ∂) implies the Kontsevich-Duflo theorem for complex manifolds.

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Section: Furthermore We Provesupporting

confidence: 56%

Section: Kontsevich-duflo Type Isomorphisms For Integrable Distributionsmentioning

confidence: 93%

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Hochschild cohomology of dg manifolds associated to integrable distributions

Chen,

Xiang,

2021

Preprint

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“…Along the way, the authors define the comma-double Lie algebroid defined by a Lie algebroid morphism, which is an interesting structure in its own right. The dg-Lie algebroid defined [27] by the comma double Lie algebroid associated to a Lie pair A ã Ñ L is used by Stiénon, Vitagliano and Xu [26] in their extension to arbitrary Lie pairs of the Kontsevich-Duflo type theorem for matched pairs in [14].…”

Section: γ G H Mmentioning

confidence: 99%

Transitive double Lie algebroids via core diagrams

Lean¹,

Mackenzie²

2021

JGM

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The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.

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“…Along the way, the authors define the commadouble Lie algebroid defined by a Lie algebroid morphism, which is an interesting structure in its own right. The dg-Lie algebroid defined [27] by the comma double Lie algebroid associated to a Lie pair A ãÑ L is used by Stiénon, Vitagliano and Xu [26] in their extension to arbitrary Lie pairs of the Kontsevich-Duflo type theorem for matched pairs in [14].…”

Section: K G H M Tg|k Th |Kmentioning

confidence: 99%

Transitive double Lie algebroids via core diagrams

Lean¹,

Mackenzie²

2021

Preprint

View full text Add to dashboard Cite

The core diagram of a double Lie algebroid consists in the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.This research was a joint project with the sadly deceased second author. This paper is dedicated to his memory.

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Formality and Kontsevich–Duflo type theorems for Lie pairs

Cited by 14 publications

References 68 publications

Hochschild cohomology of dg manifolds associated to integrable distributions

Hochschild cohomology of dg manifolds associated to integrable distributions

Transitive double Lie algebroids via core diagrams

Transitive double Lie algebroids via core diagrams

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