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 H_CE^•(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.