2023/2024 (Autumn) Topics in Mathematical Science V
Course themeFrom quiver representations to quasi-hereditary
algebras
Preliminary lecture plan
| Week | Date | Topic |
|---|---|---|
| 1 | 10-05 | Overview. Reminder: modules, representations, simple, indecomposable |
| 2 | 10-12 | Krull-Schmidt property, quiver representations, path algebras |
| 3 | 10-19 | Idempotents, composition series, Jordan-Holder theorem |
| 4 | 10-26 | Radical and socle, Artin-Wedderburn Theorem, bounded quivers |
| 5 | 11-02 | Tensor product and Hom-functor |
| 6 | 11-09 | Tensor-Hom adjunction, exactness |
| 7 | 11-16 | Projective, injective, resolution |
| 8 | 11-23 | No Lecture |
| 9 | 11-30 | Ext-group, extension, homological dimension |
| 10 | 12-07 | (Quasi-)hereditary algebras (ring theoretic definition) |
| 11 | 12-14 | Highest weight category |
| 12 | 12-21 | No lecture |
| 13 | 12-28 | No Lecture |
| 14 | 01-04 | No Lecture |
| 15 | 01-11 | Revision; properties of qha |
| 16 | 01-18 | properties of qha continue |
| 17 | 01-25 | (TBC) Non-commutative resolution via Auslander-Dlab-Ringel construction |
Time and Venue Thursday 13:00–14:30, Grad. School of Mathematics Room 409
Evaluation
- There is no examination. Grading is evaluated through homework assignments.
- In each assignment, the highest scoring question will be recorded. (->4 recorded marks)
- The next four highest scoring questions among all four assignments will also be recorded (->4 more recorded marks)
- Final grading is determined by the average of these 8 recorded marks.
- Grading scheme is as follows.
- A : 85 ∼ 100%
- B : 70 ∼ 84%
- C : 50 ∼ 69%
- Fail : 0 ∼ 49%
References
- Quiver representation
- Assem-Simson-Skowronski - Elements of the representation theory of associative al- gebras vol. 1
- D. J. Benson: Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press 1998
- K. Erdmann and T. Holm: Algebras and representation theory. Springer Undergrad- uate Mathematics Series, Springer International Publishing, 2018
- H. Derksen and J. Weyman: An introduction to quiver representations, Gradate Studies in Mathematics 184, AMS, 2017
- A. Zimmermann: Representation Theory: A Homological Algebra Point of View, Algebra and Applications 19, Springer 2014
- Homological algebra
- All the reference book for quiver representations will be useful
- H. Krause: Homological theory of representations
- J. J. Rotman: An introduction to homological algebra, second edition. Springer
- Quasi-hereditary algebra
- M. Klucznik and S. Koenig: (Lecture notes) Characteristic tilting modules over quasi- hereditary algebras
- H. Krause: Homological theory of representations