Lie Groups
A.Y. 2019/2020
Learning objectives
The course aims at providing the basic notions of Lie Groups and Lie Algebras.
Expected learning outcomes
The expected learning outcomes regard the knowledge and the ability to use Lie groups and their fundamental topological and differential properties.
Lesson period: Second semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Lie Groups
Vector Fields and associated flows. Involutive and integrable distributions, Frobenius Theorem. Lie Groups: definitions and examples, coverings of Lie Groups, fundamental groups of Lie groups. Lie algebras, invariant vector fields. Structure theorems. Adjoint representations, Killing forms, abelian Lie Groups and classification of simple compact Lie Groups.
Actions of Lie Groups
Proper and free actions. Actions of compact Lie groups. Unimodular groups. Hear measure, Slice theorem, Classification of the orbits. Symplectic manifolds, fibre bundle and associated bundles. Hamiltonian actions, moment maps. Symplectic reductions. Marsden Weistein theorem. Symplectic tori manifolds. Delzant Theorem.
Vector Fields and associated flows. Involutive and integrable distributions, Frobenius Theorem. Lie Groups: definitions and examples, coverings of Lie Groups, fundamental groups of Lie groups. Lie algebras, invariant vector fields. Structure theorems. Adjoint representations, Killing forms, abelian Lie Groups and classification of simple compact Lie Groups.
Actions of Lie Groups
Proper and free actions. Actions of compact Lie groups. Unimodular groups. Hear measure, Slice theorem, Classification of the orbits. Symplectic manifolds, fibre bundle and associated bundles. Hamiltonian actions, moment maps. Symplectic reductions. Marsden Weistein theorem. Symplectic tori manifolds. Delzant Theorem.
Prerequisites for admission
Coverings. Different results of Geometry 4
Teaching methods
Frontal lessons, in which we give examples and exercises completing the material.
Teaching Resources
M. Alexandrino R. Bettiol: Lie Groups and Geometric aspects of isometric Actions
Warner: Foundations of differentiable manifolds and Lie groups
Gallot la Fontaine Hulin "Riemannian Geometry"
Kobayashi Nomizu "Foundation of Differential Geometry I e II"
Brocker-Tom Dieck "Representations of compact Lie groups"
Warner: Foundations of differentiable manifolds and Lie groups
Gallot la Fontaine Hulin "Riemannian Geometry"
Kobayashi Nomizu "Foundation of Differential Geometry I e II"
Brocker-Tom Dieck "Representations of compact Lie groups"
Assessment methods and Criteria
The final examination consists of an oral exam.