A∞-algebras from Lie pairs

Given an inclusion A↪L of Lie algebroids sharing the same base manifold M , i.e. a Lie pair, we prove that the space Γ(Λ•A∨)⊗RU(L)U(L)⋅Γ(A), where R=C∞(M), admits an A∞-algebra structure, unique up to A∞-isomorphisms. As a consequence, the Chevalley–Eilenberg cohomology HCE•(A,U(L)U(L)⋅Γ(A)) admits a canonical associative algebra structure. This A∞-algebra can be considered as the universal enveloping algebra of the L∞-algebroid A[1]×ML/A. Our construction is based on the homotopy equivalence of the L∞-algebroid A[1]×ML/A and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz–Mackenzie.