TY - JOUR

T1 - Shifted Derived Poisson Manifolds Associated with Lie Pairs

AU - Bandiera, Ruggero

AU - Chen, Zhuo

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information:
We would like to thank Martin Bordemann, Oliver Elchinger, Camille Laurent-Gengoux, Matthew Peddie, Pavol ?evera, Jim Stasheff, and Luca Vitagliano for fruitful discussions and useful comments.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - 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 totΩA∙(Λ∙(L/A)) admits a degree (+ 1) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley–Eilenberg differential dABott:ΩA∙(Λ∙(L/A))→ΩA∙+1(Λ∙(L/A)) as unary L∞ bracket. This degree (+ 1) derived Poisson algebra structure on totΩA∙(Λ∙(L/A)) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley–Eilenberg hypercohomology H(totΩA∙(Λ∙(L/A)),dABott) admits a canonical Gerstenhaber algebra structure.

AB - 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 totΩA∙(Λ∙(L/A)) admits a degree (+ 1) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley–Eilenberg differential dABott:ΩA∙(Λ∙(L/A))→ΩA∙+1(Λ∙(L/A)) as unary L∞ bracket. This degree (+ 1) derived Poisson algebra structure on totΩA∙(Λ∙(L/A)) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley–Eilenberg hypercohomology H(totΩA∙(Λ∙(L/A)),dABott) admits a canonical Gerstenhaber algebra structure.

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U2 - 10.1007/s00220-019-03457-w

DO - 10.1007/s00220-019-03457-w

M3 - Article

AN - SCOPUS:85067314273

VL - 375

SP - 1717

EP - 1760

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -