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, given such an SH Lie pair (L, A), and any A-module E, there associates a canonical cohomology class, the Atiyah class [α E ], which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α L/A ] induces a graded Lie algebra structure on H • CE (A, L/A[−2]), and the Atiyah class [α E ] of any A-module E induces a Lie algebra module structure on H • CE (A, E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

“…( ). A similar result holds for L ∞ algebra pairs [7]. However, it is not the case in general (see an example below).…”

The Atiyah class of generalized holomorphic vector bundles

“…Proposition 2.11. Let (M, J ) be a generalized complex manifold with J given by (5) and let L − be the −i-eigenbundle of J . For a function f : U ⊂ M → , the following statements are equivalent:…”

“…Then we check that ϕ ij being a generalized holomorphic homeomorphism is equivalent to the conditions (1) and (2) as desired. Denote by (V, J 0 ) = r and let J be the generalized complex structure on M as given by (5). Then the generalized complex structure J on U ij × V is expressed as…”

“…Moreover, the Atiyah class of commutative dg algebras and the twisted version were studied in [6]. We also refer to [5,14,19] for other versions of Atiyah classes in different settings.…”

Atiyah classes and dg-Lie algebroids for matched pairs

“…From the present intermediate step, we plan, in a future paper, to construct the Atiyah class of Lie ∞-algebroid pairs (whose representations are by definition up to homotopy). This construction should naturally reduce to that given for L ∞ -algebra pairs [Chen et al, 2019] when the Lie ∞-algebroid pairs are over a point. Schematically we have the following steps towards the Atiyah class of singular foliations: L ∞ -algebra pairs [Chen et al, 2019] Lie algebroid pairs [Chen et al, 2016] Lie algebroid pairs & reps. up to homotopy (this paper)…”