Shifted derived Poisson manifolds associated with Lie pairs

TL;DR

This paper explores the connection between $L_$ algebroids and shifted derived Poisson manifolds, establishing new algebraic structures and proving a homotopy transfer theorem with applications to Lie pairs.

Contribution

It introduces a natural construction of shifted derived Poisson structures from $L_$ algebroids and proves a homotopy transfer theorem for derived Poisson algebras.

Findings

Any $L_$ algebroid induces a shifted derived Poisson manifold.

The space $ ext{tot}\u03a9_A^ullet(\u03bb^ullet(L/A))$ admits a degree (+1) derived Poisson algebra structure.

The hypercohomology $_A^{ ext{Bott}}$-cohomology has a canonical Gerstenhaber algebra structure.

Abstract

We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ 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 $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ admits a degree $(+1)$ derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential $d_A^{\operatorname{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\to \Omega^{\bullet +1}_A(\Lambda^\bullet(L/A))$ as unary $L_\infty$ bracket. This degree $(+1)$ derived Poisson algebra structure on $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology $\mathbb{H}(\Omega^{\bullet}_A(\Lambda^\bullet(L/A)),d_A^{\operatorname{Bott}})$ admits a canonical Gerstenhaber algebra structure.