Atiyah Classes of Strongly Homotopy Lie Pairs

Abstract

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-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, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [αE], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [αL/A] induces a graded Lie algebra structure on [Formula: see text], and the Atiyah class [αE] of any A-module E induces a Lie algebra module structure on [Formula: see text]. Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.