Math 595 – Lie Groupoids and Lie Algebroids – Fall 2021

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:

  • algebraic geometry: Grothendieck introduced stacks in the late 1960’s via fibered categories over a site. Fibered categories can be viewed as a type of sheaf of groupoids. More recently, this has led to the concept of a gerbe.
  • foliation theory: Haefliger introduced transversal structures to foliations in the 1970’s, using the concept of a holonomy groupoid. This approach allows for a systematic study of transversal structures, and has been central to the subsequent development of the subject.
  • noncommutative geometry and index theory: Lie groupoids made their appearance in noncommutative geometry through the monumental work of Connes in the 1980’s. He introduced the tangent groupoid of a space as a central ingredient in his approach to the Atiyah-Singer index theorem. This approach led to a number of refinements of the index theorem, such as the Connes-Skandalis index theory for foliations.
  • Poisson geometry: motivated by quantization problems, Karasev and Weinstein introduced the symplectic groupoid of a Poisson manifold in the late 1980’s, as a way to “untwist” the complicated behavior of the symplectic foliation underlying the Poisson manifold.
  • M. Crainic and R.L. Fernandes, Lectures on Integrability of Lie Brackets also available as Geometry & Topology Monographs 17 (2011) 1-107.

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 theory at the level of Math 522 – Lie groups and Lie algebras is recommended but not strictly necessary.

Email: ruiloja (at) illinois.edu
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:

  • Announcements
  • Syllabus
  • Textbooks
  • Grade Policy
  • Lecture Notes
  • Emergency information for students in Mathematics courses

Announcements:

  • Class will meet for the first time on Tuesday, August 24.
  • Lie groupoids
  • Lie algebroids
  • Lie functor and integrability
  • Differentiable stacks
  • Special Topics: To be chosen from the interests of the students.

Syllabus:

Textbooks:

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

Grading Policy

  • Expository Paper: Students will be encouraged to write (in LaTeX) and present a paper. This is not mandatory. Following the tradition of topics courses, there will be no homework and no written exams.

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.)

Emergency information for students in Mathematics courses

For important emergency information related to fires, tornados or active threats, please look at the following leaflet.

Last updated December 6, 2021