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 ext...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 [α^E], which generalizes the 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 HCE·(A,L/A[-2]), and the Atiyah class [α^E] of any A-module E induces a Lie algebra module structure on HCE(A,E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.展开更多
文摘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 [α^E], which generalizes the 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 HCE·(A,L/A[-2]), and the Atiyah class [α^E] of any A-module E induces a Lie algebra module structure on HCE(A,E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.
