Poincar\'e--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds

Abstract

We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over the same base manifold $M$ corresponds a Kapranov dg-manifold structure on $A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence, $\Gamma(\Lambda^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$ algebra structure whose unary bracket is the Chevalley--Eilenberg differential 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 $C^\infty(M)$-coalgebras $\Gamma\big(S(L/A)\big)\xrightarrow{\sim}\frac{\mathcal{U}(L)}{\mathcal{U}(L)\Gamma(A)}$, which we elect to call Poincar\'e--Birkhoff--Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map $S(\mathfrak{g})\to\mathcal{U}(\mathfrak{g})$ of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold $A[1]\oplus L/A$ is linearizable if and only if the Atiyah class of the Lie pair $(L,A)$ vanishes.