6 EC
Semester 1, period 1, 2
5334QUIV6Y
| Owner | Master Mathematics |
| Coordinator | prof. dr. Jasper Stokman |
| Part of | Master Mathematics, year 1Master Mathematical Physics, year 1 |
The goal of the course is to introduce basic techniques from representation theory, homological algebra and Lie theory and apply them to quiver representations.
A quiver is an oriented graph. A representation of a quiver is the assignment of a vector space to each vertex and the assignment of a linear map to each directed edge, with the domain and codomain of the linear map being the vector spaces attached to the source and target vertex of the directed edge. So roughly speaking, a quiver representation is a finite collection of matrices with particular size restrictions, naturally organized by an oriented graph.
Quivers nowadays play an important role in representation theory, geometry and Lie theory. The emphasis of the course is to introduce basic techniques and concepts from representation theory and homological algebra and apply them to quiver representations. Many of these techniques and concepts will reappear in master courses within the track Algebra, Geometry and Mathematical Physics. First year master students within this track are strongly encouraged to follow this course.
Topics that will be discussed are: abelian categories, indecomposable projective and injective modules, modules over path-algebras, projective and injective resolutions, ext groups, semisimple algebras and modules, bounded quiver algebras, and Gabriel’s theorem on the classification of quivers of finite representation type.
Ralf Schiffler, "Quiver representations", CMS Books in Mathematics, Springer.
Additional text on Semisimple modules and bounded quiver algebras
When the course is successfully completed the student
* has learned basic techniques from representation theory and homological algebra
* can apply these techniques to construct and decompose representations of quivers and of path algebras * can describe the indecomposable representations of Dynkin quivers
There will be a weekly meeting of 2 hours. A substantial part will be lectures, but we will also occasionally discuss and/or work on exercises. Outside the lectures the student is expected to practice by making homework exercises and recommended exercises and by studying the material that has been treated during the lectures. Occasionally the student will be asked to study a small part from the book by him/herself.
|
Activity |
Number of hours |
|
Attending lectures Individual study Working on (homework) exercises |
28 80 60 |
The programme does not have requirements concerning attendance (OER-B).
| Item and weight | Details | Remarks |
|
Final grade | ||
|
0.8 (100%) Tentamen | Must be ≥ 5.5, Allows retake | Homework exercises contribute to final mark by 20% if the mark for the exam is at least 5.5 |
Homework exercises are discarded if the mark for the exam is less than 5.5. Homework exercises are also discarded when the student takes the re-exam.
individual, contributes 20% to final mark if mark exam is at least 5.5, will be corrected and returned to the students during the lecture.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.uva.nl/plagiarism
| Weeknummer | Onderwerpen | Studiestof |
| 1 | Definition and examples of quiver representations and path algebras | Chapter 1 Section 1.1 and Chapter 4 Section 4.2, 4.3 |
| 2 | Abelian categories | Chapter 1 up to and including Section 1.3 |
| 3 | Exact sequences and Hom-functors | Chapter 1 |
| 4 | Projective and injective representations | Chapter 2 Section 2.1 |
| 5 | Krull-Remak-Schmidt theorem and standard projective resolutions | Chapter 2 Section 2.2 |
| 6 | Ext-groups | Chapter 2, Section 2.4 |
| 7 | The radical of a ring, Jacobson's theorem | Chapter 4 and the syllabus |
| 8 | ||
| 9 | Bounded quiver algebras | Chapter 5 up to and including Section 5.3 and the syllabus |
| 10 | Idempotents and connectedness of algebras | Syllabus |
| 11 | Presentations of basic algebras as bounded quiver algebras. | Syllabus |
| 12 | Morita equivalence, induction and restriction functors | Syllabus |
| 13 | The quadratic form associated to a quiver, Euclidean and Dynkin quivers | Chapter 8, Section 8.2 |
| 14 | Root systems, bricks and formulation of Gabriel's theorem | Chapter 8, Section 8.3 |
| 15 | Proof of Gabriel's theorem | Chapter 8 |
| 16 |
The schedule for this course is published on DataNose.