Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds

Abstract

We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ•A∨⊗L/A) carries a canonical L∞[1] algebra structure whose unary bracket is the Chevalley–Eilenberg differential dA∇Bott corresponding to the Bott representation of A on L/A and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair (L,A). To this end, we construct explicit isomorphisms of C∞(M)-coalgebras Γ(S(L/A))→∼U(L)U(L)Γ(A), which we elect to call Poincaré–Birkhoff–Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map S(g)→U(g) of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold A[1]⊕L/A is linearizable if and only if the Atiyah class of the Lie pair (L,A) vanishes.