Atiyah classes of strongly homotopy Lie pairs

TL;DR

This paper introduces Atiyah classes for strongly homotopy Lie pairs, revealing their role in measuring nontrivial extensions and inducing graded Lie algebra structures on cohomology, with invariance under certain deformations.

Contribution

It defines Atiyah classes for SH Lie pairs and demonstrates their influence on cohomology and deformation invariance, extending classical concepts to homotopy Lie algebra contexts.

Findings

Atiyah class induces a graded Lie algebra structure on cohomology.

Atiyah class of modules induces a Lie algebra module structure.

Invariance of Atiyah classes under gauge equivalent deformations.

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 $L$.