Sommersemester 2023

Dozent: Dr. Leonardo Patimo
Assistent: Giovanni Zaccanelli

Lectures: Thursday 16-18 in SR 404, Ernst-Zermelo-Straße 1
Exercise classes: Wednesday 14-16 in SR 218. Ernst-Zermelo-Straße 1

Content

Lie theory is a fascinating field that lies at the intersection of algebra and geometry. A Lie group is a smooth manifold with a group structure, where the group operations are smooth. Lie groups have a natural connection to geometric objects since they arise as symmetries of such objects. Examples of Lie groups include the general linear group and the orthogonal group . Moreover, the tangent space of a Lie group is naturally equipped with another important algebraic structure, known as Lie algebra.

In this lecture course, we will give an introduction to Lie groups and Lie algebras and discuss the correspondence between them. The focus of the course will be on compact Lie groups, an important class of Lie groups for which the theory is very rich and well-developed. We will then study and classify representations of compact Lie groups, that is smooth linear actions on vector spaces. As a concrete final goal, we will classify compact Lie groups in terms of more elementary data: root systems.

Lectures

S§ = Prof. Soergel's Skript

N§ = Notes on Haar Measures and the Peter-Weyl theorem.

Sep§ = Sepanski - Compact Lie groups. Lecture 1: 20.04. S§ 1.1. Groups and topological groups. Representations of groups. Classification of the complex continuous representations of S^1.
Lecture 2: 27.04. S§ 1.2. Embedded Differential manifolds. Matrix Groups. The exponential is a local chart. Matrix groups are Lie groups.
Lecture 3: 04.05. S§ 1.3, 1.4 (except 1.4.1-1.4.3), 2.1, 2.2 until 2.2.12 (except "Verschmelzungen"). Lie algebras of (matrix) Lie groups. The differential of a morphism of groups is a morphisms of Lie algebras. The differential of a representation of G is a representation of Lie(G). Irreducible representations of G are also irreducible for Lie(G).
Lecture 4: 11.05. S§ 2.2-2.3 from 2.2.13 to 2.3.5. Discussion without proofs on simply connected groups (Satz 4.3.9). Complexification of Lie algebras. Classification of the complex irreducible representations of SU(2,C) and sl(2,C).
Lecture 5: 25.05. N§ 1,2 Haar measure. Existence and uniqueness of (positive continuous density) Haar measure on Lie groups. Haar measure on compact groups and Maschke's theorem.
Lecture 6: 15.06. N§ 3,5. Matrix coefficients and their orthogonality (expept proof of Prop 3.6(2) and that isometry statement in Prop 3.9). Statement of spectral theorem. Convolutions on compact groups. (Prop 5.3 and Theorem 5.4 without proofs)
Lecture 7: 22.06. N§ 6. Proof of the Peter-Weyl theorem. Compact Lie groups have a faithful representation. S§ 4.2 Compact Lie algebras. Correspondence between compact Lie algebras and compact Lie groups with trivial center (first part of the proof.)
Lecture 8: 29.06. End of the proof of the correspondence for compact Lie algebras. Sep§ 5.1.1-5.1.3. Maximal tori and Cartan subalgebras. Compact quotients of R^n by a discrete groups are tori. Conjugation theorems for Cartan subalgebras. (see also these Section 2 of these notes by Wang

Notes

Lecture 9: 06.07. Sep§ 5.1.4. Maximal torus theorem: every element is conjugated to an element of the torus. S§ 5.2 Classification of compact groups of rank one. S§ 5.3 Weyl groups are finite groups.
Lecture 10: 13.07. S§ 5.4-5.5 Lattice reflection groups. Root systems. Maximal tori and Weyl group under surjective morphisms. Up to Proposition 5.5.14(point 1)

Notes

Notes on Haar Measures and the Peter-Weyl theorem.

Literature

Exercises

N.DateExercises
126.04Exercise Sheet 1
203.05Exercise Sheet 2
310.05Exercise Sheet 3
417.05Exercise Sheet 4
507.06Exercise Sheet 5
621.06Exercise Sheet 6
728.06Exercise Sheet 7
705.07Exercise Sheet 8
712.07Exercise Sheet 9
719.07Exercise Sheet 10

Prüfungsleistungen

Mündliche Prüfung (Dauer nach Maßgabe der Prüfungsordnung).

Studienleistungen

Mindestens 50% der erreichbaren Punkte auf die schriftlich zu bearbeitenden Übungsaufgaben.