Atiyah classes of strongly homotopy Lie pairs

Abstract

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_{\infty }$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L,A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[{\alpha }^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[{\alpha }^{L/A}]$ induces a graded Lie algebra structure on $H_{\mathrm{CE}}^{\bullet }(A,L/A[-2])$, and the Atiyah class $[{\alpha }^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $H_{\mathrm{CE}}^{\bullet }(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

30 pages; final version in Algebra Colloquium

References in corpus (5)

• The Atiyah class of a dg-vector bundle
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• The BV formalism for L${}_{\infty }$-algebras
• Invariant connections and PBW theorem for Lie groupoid pairs
• A Hopf algebra associated to a Lie pair

Cited by in corpus (1)

• The Atiyah class of generalized holomorphic vector bundles