# Lie groupoids and applications

## (Graduate Course at UdeA - Medellin)

### Why Lie groupoids?

Lie groupoids appeared first in Ehresmann's global formulation of PDE's and nowadays play an important role in Differential Geometry, Foliation Theory, Poisson Geometry, Noncommutative Geometry, Index Theory, Orbifolds and Stacks and Generalized Geometry, among others. The infinitesimal counterpart of a Lie groupoid is given by a Lie algebroid, an object that unifies Lie algebras, regular foliations but also provides an appropriate setting to study singular geometric structures, e.g. Poisson structures.

### When and Where?

Every Tuesday and Thursday from 10am to 12pm. Room 5-118.

### Office Hours

Every Monday from 10:00 to 11:00 at Room 4-331A. Students are very welcome to ask questions about any subject related to the course. Please feel free to ask your questions, I will do my best to discuss all issues that might help us solve your doubts, I will be happy to do it. If you would like to talk about Math in general, I am also available and looking forward to discuss.

### Pre-requisites

Differentiable manifolds.

### Contents

This course is an introduction to Lie groupoids and Lie algebroids, paying special attention to geometric structures on Lie groupoids and their connections with Symplectic Geometry. The course is organized as follows:

*Lie groupoids and Lie algebroids:*Lie groupoids, main concepts and examples; Lie algebroids, main concepts and examples; the Lie functor and the integrability problem.*Vector bundles over Lie groupoids and Lie algebroids:*Vector bundles over Lie groupoids (VB-groupoids), connections with representations up to homotopy, infinitesimal description (VB-algebroids).*Geometric structures on Lie groupoids:*Symplectic groupoids and Poisson structures; Riemannian groupoids; Linearization Theorem.*Lie groupoids and singular spaces:*Morita equivalence; Lie groupoids and orbifolds; Lie groupoids and quotient stacks; Symplectic groupoids vs Symplectic structures on quotient stacks.

### Notes

I will provide notes containing an expanded version of the material presented during the lectures. These lecture notes are very much in progress and the file will be updated every week. Students can require access to the lecture notes here.

### Homeworks

- Homework 1 can be found here.

### References

The following are notes and books that I recommend as very good sources of study for the first part of the course:

- M. Cranic and R. Fernandes,
*"Lectures on the integrability of Lie brackets"*. Available here. - K. Mackenzie,
*"General theory of Lie groupoids and Lie algebroids",*London Math. Soc. Lecture Notes Series 213. - I. Moerdijk and J. Mcrun,
*"Introduction to Foliations and Lie Groupoids"*.

For the second part of the course, I will follow the papers:

- Gracia-Saz, A., Mehta, R.,
*“Lie algebroid structures on double vector bundles and representation theory of Lie algebroids”,*Advances in Mathematics 223, 4, 1236-1275 (2010). Available here. - Gracia-Saz, A., Mehta, R.,
*“VB-groupoids and representation theory of groupoids”,*Journal of Symplectic Geometry**15**(3) 741-783 (2017). Available here.

but also you are encouraged to read

3. Arias Abad, C., Crainic, M., "Representations to homotopy of groupoids and the Bott spectral sequence", Adv. Math., **248**, 416-452 (2013).

4. Arias Abad, C., Crainic, M., "Representations up to homotopy of Lie algebroids", Crelle, **2012 **(663), 91-126 (2012)

where general representations up to homotopy are introduced and studied in detail.

For the third part of the course, the following references are recommended:

- J.P. Dufour and N.T. Zung,
*"Poisson structures and their normal forms"*, Progress in Mathematics, Birkhauser - M. del Hoyo and R. Fernandes,
*"Riemannian metrics on Lie groupoids",*Journal für die reine und angewandte Mathematik (Crelle) 735**,**143-173 (2018). Available here.

For the last part of the course, the following survey is highly recommended:

- M. del Hoyo,
*"Lie groupoids and differentiable stacks",*Portugaliae Mathematica, 70 (2) 161-209 (2013). Available here.