Abstract

This chapter is a brief overview of some semiclassical objects with particular relevance in Poisson geometry and deformation quantization: formal groupoids. We give a categorical description of the object, study its associated algebraic structure (Hopf algebroid), mentioning its relevance in Poisson geometry as formal realizations of Poisson manifolds.

Motivation and plan

The relation between smooth manifolds and their algebras of smooth functions has been studied deeply and the problem of connecting geometric information and algebraic data appears frequently in Lie theory and deformation theory, among others.

In particular, the notion of a groupoid appears naturally as a generalization of the structure of a group and it helps to understand geometric spaces. Its study in differential geometry allows us to link, for example, Lie groupoids and foliations of Poisson manifolds. In a more general setting, the notion of a groupoid object in a category C[1] can be introduced, and this generalized version of groupoid appears as a solution of what is called the Integrability problem or the generalized Lie Third Theorem for Lie algebroids [5] and in particular for Poisson manifolds [6], [4]. The main objective in this overview is to discuss different approaches to describe formal groupoids, which can be defined categorically as a groupoid object in a certain category, in which the properties of the object are encoded in the spaces of infinite jets associated to smooth manifolds.