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Lie groups

MATH-477

Lecturer(s) :

Raum Sven

Language:

English

Summary

This course introduces to Lie groups and their correspondence with Lie algebras. Classical groups and compact groups are treated as important classes of examples. The notions of homogeneous spaces and lattices are introduced.

Content

1. Lie groups and their Lie algebras
2. Classical groups
3. Compact (Lie) groups
4. Homogeneous spaces and lattices in Lie groups

Keywords

Lie groups, Lie group-Lie algebra correspondence, homogeneous spaces

Learning Prerequisites

Required courses

• Topology (MATH-225)
• Introduction aux varietés différentiables (MATH-322)
• Group theory (MATH-211)

Learning Outcomes

By the end of the course, the student must be able to:

• Structure the relationship between Lie groups and Lie algebras
• Define classical groups
• Describe basic properties of classical groups
• Compute the representation theory of compact Lie groups
• Explain examples of lattices and homogeneous spaces

Transversal skills

• Continue to work through difficulties or initial failure to find optimal solutions.
• Take feedback (critique) and respond in an appropriate manner.
• Demonstrate the capacity for critical thinking
• Demonstrate a capacity for creativity.
• Make an oral presentation.

Teaching methods

Ex-cathedra course with exercises

Expected student activities

• Participate in the course
• Solve regular exercises
• Present exercises in the classroom
• Acquire one short piece of mathematics independently

Assessment methods

Oral exam and presentation in the classroom. In case Art. 3 al. 5 of the regulations of the section apply to some student, the exam form will be decided by the teacher and communicated to the student.

Supervision

Office hours	Yes
Assistants	Yes
Forum	No

Resources

Bibliography

• Alexander Kirillov, Jr. An introduction to Lie groups and Lie algebras. ISBN-13: 978-0-521-88969-8
• Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces. ISBN-13: 978-0821828489
• Sigfried Echterhoff & Anton Deitmar. Principles of harmonic analysis. ISBN-13: 978-3319057910
• Theodor Bröcker & Tammo tom Dieck. Representations of compact Lie groups. ISBN-13: 978-3540136781
• David Witte Morris. Introduction to arithmetic groups. ISBN-13: 978-0986571602

Ressources en bibliothèque

• An introduction to Lie groups and Lie algebras / Kirillov
• Differential geometry, Lie groups, and symmetric spaces / Helgason
• Principles of harmonic analysis / Echterhoff & Deitmar
• (electronic version)
• Representations of compact Lie groups / Bröcker & Dieck