MS Seminar (Mathematics - String Theory)

Speaker: Alexander Voronov (University of Minnesota)
Title: Homotopy Lie algebroids and bialgebroids
Date: Thu, Apr 20, 2017, 15:30 - 17:00
Place: Seminar Room A
Abstract: Lie algebroids appear throughout geometry and mathematical physics and realize the idea of a family of Lie algebras parameterized by a smooth manifold. A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds, which might be regarded as a version of derived geometry. This leads naturally to the notion of an L_infty-algebroid, which offers an alternative way to think of a family of L_infty-algebras over a smooth manifold as compared to K. Costello's notion of an L_infty space. The situation with Lie bialgebroids and their morphisms is more complicated, as they combine covariant and contravriant features. We approach them in terms of odd symplectic dg-manifolds, building on the work of D. Roytenberg. We extend them to the homotopy Lie case and introduce the notions of an $L_\infty$-bialgebroid and an $L_\infty$-morphism between them. This is a joint work with my student Denis Bashkirov.