University of Illinois at Urbana-Champaign

Math 595 - Lie Groupoids and Lie Algebroids - Fall 2021

(section LG)

This course is an introduction to the theory of Lie groupoids and their infinitesimal counterparts, called Lie algebroids. This is a far reaching extension of the usual Lie theory, which finds application in many areas of Mathematics.

Groups typically arise as the symmetries of some given object. The concept of a groupoid allows for more general symmetries, acting on a collection of objects rather than just a single one. Groupoid elements may be pictured as arrows from a source object to a target object, and two such arrows can be composed if and only if the second arrow starts where the first arrow ends. Just as Lie groups (as introduced by Lie around 1900) describe smooth symmetries of an object, Lie groupoids (as introduced by Ehresmann in the late 1950's) describe smooth symmetries of a smooth family of objects. That is, the collection of arrows is a manifold G, the set of objects is a manifold M, and all the structure maps of the groupoid are smooth. Ehresmann's original work was motivated by applications to differential equations. Since then, Lie groupoids have appeared in many other branches of mathematics and physics. These include: In this course I will not be following any particular book, but the following Lecture Notes should be useful: I will also provide notes of the lectures.

Students taking this course are assumed to know differential geometry at the level of Math 518 - Differentiable Manifolds. A knowledge of ordinary Lie Theorey at the level of Math 522 is recommended but not strictly necessary.

Email: ruiloja (at)
Office: 366 Altgeld Hall
Office Hours: 11:30AM-12:30PM TR
Class meets: 02:00-03:20PM TR, 345 Altgeld Hall
Prerequisites: Math 518 or equivalent.

In this page:




I will provide to participants some lecture notes as the course progresses, but the following references should also be very helpful:

Grading Policy

Lecture Notes:

Part 1: General Theory

Part 2: Singular Spaces

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

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Last updated December 6, 2021