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 $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[\alpha^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.