6 ECTS credits
150 h study time
Offer 1 with catalog number 4013458FNR for all students in the 2nd semester of even academic years (e.g. 2012-2013) at a (F) Master - specialised level.
The aim is to acquire a good understanding and intuition on the topics studied. All notions and proofs should be well understood and the student should be able to communicate this in a written and oral manner. Furthermore, the student should be able to discover and prove independently related properties and structures. The student should be able to make independently a search of the necessary literature (and this also in the English language).
In representation theory one studies descriptions (representations) of groups and algebras via morphisms of the given structures in more known structures, such as direct products of matrix rings over skew fields. Via representation theory one obtains a close relationship between groups and rings, and this via the group algebra. In this course we study the algebraic structure of these algebras.
We begin with an introduction to algebras and non-commutative rings. Next we study semisimple group rings and descriptions of their Wedderburn decomposition. Our attention mainly goes to some of the following topics.
- (Complex) Representation theory of finite groups.
- Applications of representation theory, including the celebrated theorems of Burnside, Hurwitz and Frobenius.
- McKay's conjecture and some results related to solvable groups.
- Introduction to representation theory of symmetric groups and some applications.
- Rational representations and rational group algebras.
- Integral group rings and the isomorphism problem.
- The unit group of orders and integral group rings, including constructions of units, construction of generators, construction of free subgroups, structure theorems.
- Study of other types of algebras and their representation theory, such as Lie algebras and their quantum deformations.
- Study of Hopf algebras and their representation theory, braided algebras and tensor categories.
Books:
Kassel, Quantum groups, Springer.
Lam, A first course in non-commutative rings, Springer.
Jespers, del Rio, Group ring groups, vol1 and vol2, De Gruyter, Berlin, 2016.
Polcino Milies, Cesar; Sehgal, Sudarshan K., An introduction to group rings. Algebras and Applications, 1. Kluwer Academic Publishers, Dordrecht, 2002. xii+371 pp. ISBN: 1-4020-0238-6
Additional Study Material
The student should also consult other books.
1. Student knows and has insight into fundamental results in representation theory of algebras.
2. Student can look up related theory.
3. Student can analyse and understand related theory.
4. Student can make connections with other theories.
5. Student can synthesize and interpret results.
6. Student independently can and understand consult recent literature.
7. Student independently can prepare a mathematics text about another theory and report orally.
8. Student can analyze results.
9. Student independently can look up and solve exercises.
10. Student can think in function of problem solving thinking.
The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.
Within the Other Exam category, the following assignments need to be completed:
Exam: oral exam and a project. A mark will only be assigend if the student has participated in all test, exams and assignments.
The topic of the project will be discussed during the course. It can be a study of some topics related to representation theory or the study of a research article. The student should come up with a proposal within the first 4 weeks. After 8 weeks the student should present a written document showing sufficient progress. A final written document has to be submitted at the commencement of the oral exam. The exam starts with a short oral presentation of the project. The student will be evaluated on the understanding of the material and on the broader view (insight) of the topic investigated.
This offer is part of the following study plans:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)