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, $\mathrm{\Gamma }({\mathrm{\Lambda }}^{•}{A}^{\vee }\otimes L/A)$ carries a canonical $L_{\infty }[1]$ algebra structure whose unary bracket is the Chevalley–Eilenberg differential $d_A^{{\mathrm{\nabla }}^{\mathrm{Bott}}}$ 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 $C^{\infty }(M)$-coalgebras $\mathrm{\Gamma }(S(L/A))\stackrel{\sim }{\to }\frac{\mathcal{U}(L)}{\mathcal{U}(L)\mathrm{\Gamma }(A)}$, 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 $S(\mathfrak{g})\to \mathcal{U}(\mathfrak{g})$ 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.

我们证明，对于每一个包含 $\mathrm{一种}\hookrightarrow 升$相同基流形M 上的李代数对应于上的 Kapranov dg-流形结构$\mathrm{一种}[1]\oplus 升/\mathrm{一种}$，这对于同构来说是规范的。作为结果，$\mathrm{\Gamma }({\mathrm{\Lambda }}^{•}{\mathrm{一种}}^{\vee }\otimes 升/\mathrm{一种})$ 携带规范 ${升}_{\infty }[1]$ 一元括号是 Chevalley-Eilenberg 微分的代数结构 $d_{\mathrm{一种}}^{{\mathrm{\nabla }}^{\mathrm{博特}}}$对应于的博特表示甲上$升/\mathrm{一种}$ 其二元括号是李对的 Atiyah 类的共环代表 $(升,\mathrm{一种})$. 为此，我们构造了显式的同构$C^{\infty }(米)$-代数 $\mathrm{\Gamma }(秒(升/\mathrm{一种}))\stackrel{～}{\to }\frac{\mathcal{你}(升)}{\mathcal{你}(升)\mathrm{\Gamma }(\mathrm{一种})}$，我们选择将其称为 Poincaré-Birkhoff-Witt 映射。这些映射允许递归表征，允许显式计算。他们概括了经典的对称化映射$秒(\mathfrak{G})\to \mathcal{你}(\mathfrak{G})$Lie 理论和（的逆）微分算子的完整符号映射。最后，我们证明 Kapranov dg-流形$\mathrm{一种}[1]\oplus 升/\mathrm{一种}$ 可线性化当且仅当 Lie 对的 Atiyah 类 $(升,\mathrm{一种})$ 消失。