Shifted derived Poisson manifolds associated with Lie pairs

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