Representation theory
MATH-334 / 5 credits
Teacher:
Language: English
Remark: pas donné en 2023-24
Summary
Study the basics of representation theory of groups and associative algebras.
Content
Representations of associative algebras in complex vector spaces. General results: subrepresentations and semisimple representations. Modules over rings and associative algebras. Homomorphisms of modules. Wedderburn's theorem.
Examples of algebras, group algebras of finite groups, quiver algebras and universal enveloping of Lie algebras. Representations of finite groups. Maschke's theorem. Tensor products.
The density theorem. Characters. The Jordan-Hölder and Krull-Schmidt theorems. Extensions of representations.
Keywords
Linear representation, subrepresentation, quotient, simple representation, subrepresentation, quotient representation, semisimple representation, character of a representation, associative algebra, representation of a finite group.
Learning Prerequisites
Required courses
Linear algebra or Advanced Linear algebra; Group theory
Recommended courses
Lie algebras, Coxeter groups
Learning Outcomes
By the end of the course, the student must be able to:
- Apply concepts and ideas of the course
- Reason rigorously using the notions of the course
- Choose an appropriate method to solve problems
- Identify the concepts relevant to each problem
- Apply concepts to solve problems similar to the questions in problem sets
- Solve new problems using the ideas of the course
- Implement appropriate methods to study and construct representations of groups and algebras
Teaching methods
Lectures and exercise sessions
Assessment methods
One take-home written assignment (15% of the grade)
Written exam (85% of the grade)
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | No |
Resources
Bibliography
1. P. Etingof, O. Goldberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, "Introduction to Representation Theory". Student Mathematical Library Volume: 59; 2011. ISBN: 978-0-8218-5351-1
2. Fulton, William, and Joe Harris. Representation Theory: A First Course. Graduate texts in mathematics. Vol. 129. New York, NY: Springer, 1991. ISBN: 9780387974958.
Ressources en bibliothèque
Moodle Link
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Representation theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
Reference week
| Mo | Tu | We | Th | Fr | |
| 8-9 | |||||
| 9-10 | |||||
| 10-11 | |||||
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| 20-21 | |||||
| 21-22 |
Légendes:
Lecture
Exercise, TP
Project, other