Poincaré--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 $\Gamma (S(L/A))\stackrel{\sim }{\to }\frac{\mathcal{U}(L)}{\mathcal{U}(L)\Gamma (A)}$, 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.

48 pages

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Cited by in corpus (14)

• The Atiyah class of a dg-vector bundle
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Invariant connections and PBW theorem for Lie groupoid pairs
• Various instances of Harish-Chandra pairs
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Polyvector fields and polydifferential operators associated with Lie pairs
• An index theorem for Lie algebroids
• The geometric constraints on Filippov algebroids
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• $A_{\infty }$-Algebras from Lie Pairs
• Dg Loday-Pirashvili modules over Lie algebras
• The Strong Homotopy Structure of BRST Reduction