Mini course: An Introduction to Lie Groupoids, by René Langøen

Abstract: 

Lie groupoids generalize Lie groups by allowing only partial multiplication—not every pair of elements can be multiplied—thus capturing symmetries that vary across a space. Fundamental examples arise from Lie group actions on manifolds. In the case of a transitive action, this provides an alternative to the classical construction of a homogeneous manifold, such as a principal H-bundle, where H is the isotropy subgroup at a chosen point. The groupoid perspective offers a more intrinsic framework, avoiding the need to select a reference point. Another example is given by an action of a Lie group on a manifold, a so-called action groupoid. 

In this seminar series, I will introduce the general definition of a Lie groupoid and present several illustrative examples. We will compare the structure of Lie groupoids with that of Lie groups, and explore how the construction of a Lie algebra associated with a Lie group can be generalized to relate a Lie algebroid to any Lie groupoid.

The remainder of the course will aim to demonstrate how various notions of curvature can be naturally unified within the framework of Lie algebroids.