Lie groupoids and algebroids

Garant předmětu:

Jan Vysoký

Lecturer:

Jan Vysoký

Tutor:

Jan Vysoký

Supervisor:

Department of Physics

Synopsis:

Lie groups and algebras are undoubtedly the cornerstone of modern theoretical physics and differential geometry. It turns out that in certain situations (point-dependent symmetries, non-transitive actions of Lie groups), it is appropriate to extend this concept.

Groupoids are a natural generalization of groups. In simplified terms, group elements are replaced by „arrows“ that can be associatively multiplied only when they „connect“ in a certain sense. An illustrative example is the transition from the group of homotopy classes of loops at a given point to the groupoid of homotopy classes of general curves. If all the involved sets are manifolds and the corresponding operations are smooth, we speak of Lie groupoids. As the name suggests, Lie algebroids are the corresponding „infinitesimal objects.“ Mathematically, these are vector bundles, whose module of smooth sections forms a Lie algebra.

In this lecture, we will explore basic concepts and constructions, with a focus on various examples of Lie groupoids/algebroids across differential geometry. We will also discuss their significance in Poisson and symplectic geometry.

Students are expected to have a good understanding of basic concepts in differential geometry, especially in Lie theory and the theory of principal bundles.

Requirements:

Syllabus of lectures:

1.Lie groupoids

2.Transitivity and local triviality

3.Bisections and actions

4.Algebraic constructions with Lie groupoids

5.Lie algebroids

6.Lie functor

7.Exponential maps, adjoint representations

8.Algebraic constructions with Lie algebroids

9.Poisson structures and Lie algebroids

10.Poisson and symplectic Lie groupoids

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] K. Mackenzie: General theory of Lie groupoids and Lie algebroids. Cambridge University Press, 2005.

[2] I. Moerdijk, J. Mrcun: Introduction to Foliations and Lie groupoids. Cambridge University Press, 2003.

Recommended references:

[3] J. Pradines: In Ehresmann‘s footsteps: from Group Geometries to Groupoid Geometries. Banach Center Publications vol. 76, 87-157, 2007.

[4] A. Weinstein: Poisson geometry. Differential geometry and its applications 9(1-2), 213-238, 2007.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans: