Homotopy Lie algebroids and bialgebroids

In these lectures I will give an introduction to homotopy Lie (or L-)algebroids and bialgebroids. I will not rely on any knowledge of ordinary, nonhomotopy Lie algebroids and bialgebroids and try to introduce necessary notions along the way.

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-algebroid, which offers an alternative way to think of a family of L-algebras over a smooth manifold as compared to K. Costello's notion of an L space.

The situation with L-bialgebroids and their morphisms is more complicated, as they combine covariant and contravriant features. I will discuss an approach to them in terms of odd symplectic dg-manifolds. These lectures will be based on classical works of Vaintrob, Kosmann-Schwarzbach, Ping Xu, and Roytenberg in the nonhomotopy case and a recent joint work with my student Denis Bashkirov regarding the L extension.

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