Representation theory I

This page contains the information about the course V4A3 Representation Theory I, taught by Prof. Dr. Catharina Stroppel.

In this course we will study complex semisimple Lie algebras, their classification and their representation theory.

Lie algebras arise naturally as tangent spaces of Lie groups or affine algebraic groups. Their representation theory is a model for many other — often more complicated and more involved — structures in representation theory. We in particular will learn the concept of (highest) weight theory and the basic concepts of reflection groups. The first part will follow standard literature like [H], [K]. The second part will be on root systems, integrality questions and Chevalley groups. If time allows we will finally deal with the famous category O of infinite dimensional highest weight modules.

Prerequisites Good knowledge of basics in algebra is absolutely necessary. Some knowledge of Lie groups or affine algebraic groups will be useful for the second part.

Lectures

Wednesday, 10–12

Zeichensaal, Wegelerstraße 10

(change of room!)

Friday, 8–10

Größer Hörsaal, Wegelstraße 10

The first lecture is April 5.

Tutorials

The first tutorial takes place in the second week of the semester (i.e. April 8–12).

• Monday, 10–12, N.003
• Wednesday, 14–16, N.008
• Thursday, 12–14, N.003

The tutors are

• Calvin Pfeifer (Monday)
• Tomasz Przezdziecki (Wednesday)
• Jendrik Stelzner (Thursday)

The group distribution is available here.

Solutions to the homework must be handed in before the lecture on Wednesday. The first time this happens is on April 10, you will get this homework after the first lecture. In the first tutorial session (i.e. in the second week of the semester) there will be problems to be solved during the tutorial.

Exercise sheets

• tutorial exercises
• sheet 1 (April 8: the definition of $\mathfrak{sp}_{2n}$ was fixed)
• sheet 2
• sheet 3 (April 21: fixed a typo in the definition of $V_\lambda$)
• sheet 4
• sheet 5 (May 8: fixed characteristic in Clebsch–Gordan decomposition)
• sheet 6
• sheet 7 (May 19: added some reminders on definitions from earlier on the course)
• sheet 8 (May 24: fixed a typo involving coroots)
• sheet 9
• sheet 10
• sheet 11 (June 24: $V$ is the standard representation)
• sheet 12
• sheet 13

Jendrik Stelzner has partial unofficial solutions written up.

Exam

• Wednesday, July 17 2019, 9–11, Großer Hörsaal
• Friday, September 20 2019, 9–11, Kleiner Hörsaal

References

[H] J, Humphreys, Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9, Springer 1978.

[K] A. Knapp, Lie groups beyond an introduction. Progress in Mathematics, 140. Birkhäuser 2002.