# Fiches de cours

## Representation theory of semisimple lie algebras

Testerman Donna

English

#### Summary

We will establish the major results in the representation theory of semisimple Lie algebras over the field of complex numbers, and that of the related algebraic groups.

#### Content

Highest weight theory

Universal enveloping algebra

Construction of irreducible representations

Weyl's degree formula

Freudenthal's formula.

If time permits, construction of Chevalley groups and simple algebraic groups.

#### Learning Prerequisites

##### Required courses

Theorie des Groupes, Anneaux et corps, Algebres de Lie semisimples

##### Important concepts to start the course

The classification of complex semisimple Lie algebras. Root systems.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Construct irreducible representations of a given highest weight
• Formulate main results on the representation theory
• Sketch proofs of some main results
• Carry out routine calculations
• Compute dimensions of weight spaces and modules

#### Transversal skills

• Assess one's own level of skill acquisition, and plan their on-going learning goals.
• Continue to work through difficulties or initial failure to find optimal solutions.
• Demonstrate the capacity for critical thinking
• Use a work methodology appropriate to the task.
• Plan and carry out activities in a way which makes optimal use of available time and other resources.

Lectures

#### Expected student activities

Exercises, extra reading, presentation of exercises.

#### Assessment methods

Final written exam

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

#### Supervision

 Office hours Yes Assistants Yes

#### Resources

##### Bibliography

James Humphreys : Introduction to Lie algebras and Representation Theory.

Bourbaki, Lie algebras and Lie groups, Chapters 1 - 3.

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand