Description: Lie groupoids are geometric objects that unify smooth manifolds, Lie groups, actions of Lie groups, among others. Also, groupoids provide a natural framework to deal with singular spaces such as orbifolds. Just as Lie groups, a Lie groupoid has an infinitesimal invariant whose structure is that of a Lie algebroid. Roughly, a Lie algebroid can be thought of as a “generalized tangent bundle” and many geometric structures can be encoded in terms of Lie algebroids, e.g. regular foliations, Poisson structures, Dirac structures, among others. Unlike finite dimensional Lie algebras, third Lie Theorem does not hold in general in the Lie algebroid setting, i.e. not every Lie algebroid comes from a Lie groupoid. If one restricts to integrable Lie algebroids, i.e. those coming from Lie groupoids, then there exists a unique integration up to covering. Usually, one is interested in the global picture of geometric structures defined via Lie algebroid data, e.g. Poisson structures, Dirac structures, so it is interesting to study what kind of geometric structure is induced on the unique integrating Lie groupoid. Geometric structures on smooth manifolds are generally defined via tensors and subbundles of tangent and cotangent bundles, so in order to deal with geometric structures on Lie groupoids one is naturally led to work with tangent bundles and cotangent bundles of Lie groupoids. These are examples of a geometric structure referred to as VB-groupoid, that is, a vector bundle object in the category of Lie groupoids.

The main goal of this course is to study the geometry of vector bundles over Lie groupoids, offering a unified setting to work with vector bundles over manifolds and equivariant vector bundles. Our main motivation comes from symplectic/Poisson geometry and more generally Dirac geometry, hence key examples will be discussed along the way. An important observation is that roughly, vector bundles over Lie groupoids correspond to linear actions of Lie groupoids, i.e. representation theory of groupoids. The course aims at understanding the geometry of the theory of VB-groupoids and its connection with representation theory of Lie groupoids, having in mind applications to Poisson geometry and Dirac structures on groupoids.

Schedule: Wednesday from 16:30 to 19:30. Lectures from 16:30 to 18:50 with a break of 20 minutes. Then, from 19:00 to 20:00 we will have a discussion session about exercises, general questions, proofs, etc. All our activities are going to take place at Room A252 Block A of IME.

Teaching Assistance: This course will not have a TA. However, we will meet once a week to discuss exercises and also to talk about general questions, proofs, etc, as indicated in the previous item.

Grading: The grades will be based on participation during the class and some assignments. Additionally, there will be a final research project about a topic related to the course. The students have to choose a topic of their interest, then write down a short survey (15 pages at most) and give a seminar about the project. A list containing suggestions for possible projects can be found here.

Program: We plan to cover the following topics

Lie groupoids: definition and examples; actions of Lie groupoids; Morita equivalence; relation with orbifolds; cohomology of Lie groupoids; relation with equivariant cohomology.

Representation theory of Lie groupoids: brief review of vector bundles on manifolds; representations of Lie groupoids, examples; cohomology with coefficients in a representation; Morita invariance of cohomology.

Vector bundles over Lie groupoids: VB-groupoids; examples; gauge transformations; classification of regular VB-groupoids; relation with representations up to homotopy; adjoint and coadjoint representations; associated cohomologies.

Infinitesimal counterparts: VB-algebroids; examples; relation with representations up to homotopy; integration of VB-algebroids; applications to Dirac structures on groupoids.

Additional topics: If time permits, we will cover some other topics, including: basic K-theory; K-theory of orbifolds; connection with noncommutative geometry; quantization schemes.

References: References 3) and 4) cover the basics on Lie groupoids and Lie algebroids. We will follow 2) to cover most of the material about vector bundles over Lie groupoids and representation theory of groupoids. Similarly, vector bundles over Lie algebroids and representation theory of Lie algebroids is treated in 1), so we will follow this reference closely.

1. 1)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)
   

2) Gracia-Saz, A., Mehta, R., “VB-groupoids and representation theory of groupoids”, arxiv:1007.3658 (2011)

3) Mackenzie, K., “General theory of Lie groupoids and Lie algebroids”, Lecture Notes London Math. Soc. 213 (2005)

4. 4)Moerdjik, I., Mrcun, J., “Introduction to foliations and Lie groupoids” Cambridge University Press (2003)
   

Assignments: The assignments contain exercises which worth to be done and which are related to the theory discussed in the lectures. Think about the exercises, talk with your colleagues and bring your comments, solutions, doubts, etc to the discussion time.

1. -Assignment1
   

2. -Assignment2