LAUSR

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