# Shifted Derived Poisson Manifolds Associated with Lie Pairs

@article{Bandiera2019ShiftedDP,
title={Shifted Derived Poisson Manifolds Associated with Lie Pairs},
author={Ruggero Bandiera and Zhuo Chen and Mathieu Sti{\'e}non and Ping Xu},
journal={Communications in Mathematical Physics},
year={2019},
volume={375},
pages={1717-1760}
}
• Published 2 December 2017
• Mathematics
• Communications in Mathematical Physics
We study the shifted analogue of the “Lie–Poisson” construction for $$L_\infty$$ L ∞ algebroids and we prove that any $$L_\infty$$ 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 \hbox {tot}\Omega ^{\bullet }_A(\Lambda…
Quantization of (-1)-Shifted Derived Poisson Manifolds
• Mathematics
• 2022
. We inv estigate the quantization problem of ( − 1) -shifted derived Poisson manifolds in terms of BV ∞ -operators on the space of Berezinian half-densities. We prove that quantizing such a ( − 1)
Polyvector fields and polydifferential operators associated with Lie pairs
• Mathematics
Journal of Noncommutative Geometry
• 2021
We prove that the spaces $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and $\operatorname{tot}\big(\Gamma(\Lambda^\bullet Internal symmetry of the$L_{\leqslant 3}$algebra arising from a Lie pair • Mathematics • 2022 : A Lie pair is an inclusion A to L of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical L 6 3 algebra Γ( ∧ • A ∨ Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras • Mathematics Communications in Mathematical Physics • 2022 This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C ∞ context. We prove that, for The standard cohomology of regular Courant algebroids • Mathematics • 2021 For any Courant algebroid E over a smooth manifold M with characteristic distribution F which is regular, we study the standard cohomology H• st(E) by using a special spectral sequence. We prove a Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids • Mathematics • 2020 In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the Dirac generating operators of split Courant algebroids • Mathematics • 2022 Given a vector bundle A over a smooth manifold M such that the square root L of the line bundle ∧A ⊗ ∧T M exists, the Clifford bundle associated to the split pseudoEuclidean vector bundle (E = A⊕A∗, Homotopy equivalence of shifted cotangent bundles Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy Hochschild cohomology of dg manifolds associated to integrable distributions • Mathematics • 2021 For the field K = R or C, and an integrable distribution F ⊆ TKM = TM ⊗R K on a smooth manifold M , we study the Hochschild cohomology of the dg manifold (F [1], dF ) and establish a canonical ## References SHOWING 1-10 OF 70 REFERENCES Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids • Mathematics • 2013 We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson Thick morphisms, higher Koszul brackets, and$L_{\infty}$-algebroids • Mathematics • 2018 It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on Deformation of Dirac Structures via L∞ Algebras • Mathematics International Mathematics Research Notices • 2018 The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not Fedosov dg manifolds associated with Lie pairs • Mathematics Mathematische Annalen • 2020 Given any pair$(L,A)$of Lie algebroids, we construct a differential graded manifold$(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the cohomological vector field$Q$Manin Triples for Lie Bialgebroids • Mathematics • 1995 In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does Polyvector fields and polydifferential operators associated with Lie pairs • Mathematics Journal of Noncommutative Geometry • 2021 We prove that the spaces$\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$and$\operatorname{tot}\big(\Gamma(\Lambda^\bullet
From Atiyah Classes to Homotopy Leibniz Algebras
• Mathematics
• 2012
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes TX[−1] into a Lie algebra object in D+ (X), the bounded below derived category of
A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras
• Mathematics
• 2014
In this paper we consider $L_{\infty}$-algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an $L_{\infty}$ quasi-isomorphism $U: L \to \tilde{L}$
Courant algebroids, derived brackets and even symplectic supermanifolds
In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant