Shifted Derived Poisson Manifolds Associated with Lie Pairs

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.