### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1717-1760 |

Number of pages | 44 |

Journal | Communications In Mathematical Physics |

Volume | 375 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2020 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*375*(3), 1717-1760. https://doi.org/10.1007/s00220-019-03457-w