# Coursebooks

## 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

In addition to a final exam, part of the grade will be based upon student presentation of some course material during the exercise sessions or corrected written homework assignments, or both.

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.

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
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20-21
21-22
Under construction

Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German