Seminar: Matrix (Lie) Groups

Summer Semester 2023

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Lecture Schedule

For a detailed overview of the talks, visit the link .

Date	Talk	Notes
April 20	Introduction to Matrix Groups and Examples Topology of Matrix Groups	♦ Handouts ♦ Example of a non-matrix group ♦ Quotient groups
April 27	Lie Algebra	♦ Handouts
May 4	Lie Bracket	♦ Handouts
May 11	Matrix Exponentiation Homogeneous Spaces	♦ Handouts
May 25	Maximal Tori Matrix Groups as Manifolds	♦ Handouts (Maximal Tori) ♦ Handouts (Manifolds)
June 15	Structure of the Adjoint Representation	♦ Handouts
June 29	Classification of Compact Matrix groups Introduction to Lie Groups: What Lies Beyond	♦ Handouts

Description

This is a seminar targeted at mid-level as well as advanced undergraduate students. The goal of the course is to introduce some basic ideas in Lie theory through matrix groups. There will be a strong emphasis on looking at plenty of examples of matrix groups.

The first part of the course will be a fairly general introduction to matrix groups and their Lie algebras - both as algebraic objects and as tangent spaces at identity to the respective groups. In the second part, we will focus on compact matrix groups and discuss their structure theory. At the end, we will cover the classification theorem of compact semi-simple Lie groups and look at some concrete examples.

Recommended Prerequisites

Linear Algebra, Analysis (Basic topology in Rn and the notion of derivatives), Basic abstract algebra (group theory). Roughly speaking, if you feel comfortable with the first three chapters of Reference (1) below, this class is well-suited for you.

Grading

You must give a talk to receive a grade. The quality of your talk will determine your grade.

References

• (A very simple introductory textbook) Matrix Groups for undergraduates, Kristopher Tapp (AMS).
• (Main Reference Text) Matrix Groups: an Introduction to Lie Group Theory, Andrew Baker, Springer, 2002.

Further reading

• Lie groups, Lie algebras, and Representation, Brian Hall, Springer
• Representation of Compact Lie groups, Theodor Bröcker and Tammo tom Dieck, Springer