Formality for g-manifolds

Abstract

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\mathrm{tot}\left({\Lambda }^{\bullet }{\mathfrak{g}}^{\vee }{\otimes }_{\mathbb{k}}{T}_{\mathrm{poly}}^{\bullet }\right)$ and $\mathrm{tot}\left({\Lambda }^{\bullet }{\mathfrak{g}}^{\vee }{\otimes }_{\mathbb{k}}{D}_{\mathrm{poly}}^{\bullet }\right)$, whose cohomologies $H_{\mathrm{CE}}(\mathfrak{g},T_{\mathrm{poly}}^{\bullet }\stackrel{0}{\to }{T}_{\mathrm{poly}}^{\bullet +1})$ and $H_{\mathrm{CE}}(\mathfrak{g},D_{\mathrm{poly}}^{\bullet }\stackrel{0}{\to }{D}_{\mathrm{poly}}^{\bullet +1})$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\Phi :\mathrm{tot}\left({\Lambda }^{\bullet }{\mathfrak{g}}^{\vee }{\otimes }_{\mathbb{k}}{T}_{\mathrm{poly}}^{\bullet }\right)\to \mathrm{tot}\left({\Lambda }^{\bullet }{\mathfrak{g}}^{\vee }{\otimes }_{\mathbb{k}}{D}_{\mathrm{poly}}^{\bullet }\right)$ whose first `Taylor coefficient' (1) is equal to the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold $M$ and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold $M$ is an isomorphism of Gerstenhaber algebras from $H_{\mathrm{CE}}(\mathfrak{g},T_{\mathrm{poly}}^{\bullet }\stackrel{0}{\to }{T}_{\mathrm{poly}}^{\bullet +1})$ to $H_{\mathrm{CE}}(\mathfrak{g},D_{\mathrm{poly}}^{\bullet }\stackrel{0}{\to }{D}_{\mathrm{poly}}^{\bullet +1})$.

8 pages. Updated references. Fix typos. To appear in Compte Rendus Math

References in corpus (1)

• The Atiyah class of a dg-vector bundle

Cited by in corpus (7)

• Shifted derived Poisson manifolds associated with Lie pairs
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Fedosov dg manifolds associated with Lie pairs
• Hopf algebras arising from dg manifolds
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Polyvector fields and polydifferential operators associated with Lie pairs
• $A_{\infty }$-Algebras from Lie Pairs