Fedosov dg manifolds associated with Lie pairs

A. We study the shifted analogue of the "Lie-Poisson" construction for L∞ algebroids and we prove that any L∞ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras.As an application, we prove that, given a Lie pair (L, A), the space tot Ω • A (Λ • (L/A)) admits a degree (+1) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential d Bott A arXiv:1712.00665v2 [math.QA]

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“…As a consequence, (M = L[1] ⊕ B, Q) is a dg manifold, called the Fedosov dg manifold [48]. Consider the surjective submersion M → M .…”

Section: Second Construction: Fedosov Dg Lie Algebroid Arising From A...mentioning

confidence: 99%

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Bandiera

Chen

Stiénon

et al. 2019

Commun. Math. Phys.

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“…By taking symmetric tensor product, its k-th symmetric tensor product S k A (V [1]⊕U ) homotopy contracts onto the k-th symmetric tensor product S k A (U/V ) of the A-module U/V (cf. [37,41]). For completeness, we will sketch a proof below.…”

Section: 21mentioning

confidence: 99%

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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“…Given a Lie pair (L, A) and having chosen some additional geometric data, one can endow the graded manifold M = L[1] ⊕ L/A with a structure of dg manifold (M, Q), called Fedosov dg manifold [32]. It turns out that there exists a natural dg integrable distribution F ⊂ T M on (M, Q).…”

Section: Questionmentioning

confidence: 99%

“…In this section, we recall the basic construction of Fedosov dg manifolds of a Lie pair. For details, see [32].…”

Section: F Lmentioning

confidence: 99%

Polyvector fields and polydifferential operators associated with Lie pairs

Bandiera,

Stiénon,

2019

Preprint

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A. We prove that the spaces tot Γ(Λ • A ∨ ) ⊗R T • poly and tot Γ(Λ • A ∨ ) ⊗R D • poly associated with a Lie pair (L, A) each carry an L∞ algebra structure canonical up to an L∞ 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), or more precisely, on the fibers of the map of differentiable stacks XA → XL associated with the Lie pair (L, A). Consequently, bothadmit 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).

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Fedosov dg manifolds associated with Lie pairs

Cited by 12 publications

References 24 publications

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Hochschild cohomology of dg manifolds associated to integrable distributions

Polyvector fields and polydifferential operators associated with Lie pairs

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