Representation theory of semisimple lie algebras
MATH-492 / 5 crédits
Enseignant:
Langue: Anglais
Remark: pas donné en 2024-25 - Cours donné en alternance tous les deux ans
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.
Teaching methods
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.
Ressources en bibliothèque
- Lie algebras and Lie groups / Bourbaki
- Introduction to Lie algebras and Representation Theory / Humphreys
Moodle Link
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Representation theory of semisimple lie algebras
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Representation theory of semisimple lie algebras
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Representation theory of semisimple lie algebras
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
Semaine de référence
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