2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra G
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hironori Oya
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon
- Class
- -
- Course Code
- MTH.A503
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
The main topic of this course is representation theory of quivers. The representation theory of quivers can be thought of as an extension of linear algebra, and can be regarded as a part of representation theory of associative algebras. In the study of representation theory of quivers, we essentially use various mathematical tools such as category theory, homological algebra, and geometry. The representation theory of quivers has applications in many fields, including the theory of cluster algebras and the representation theory of Lie algebras and quantum groups.
The aim of this course together with "Advanced topics in Algebra H" is to introduce the basic notions in the representation theory of quivers and to explain its applications. Through this course, students are expected to learn that various mathematical tools are essentially used in the representation theory.
Course description and aims
- To be able to explain the definition and examples of representations of quivers.
- To be able to explain the definition and examples of path algebras.
- To be able to state Gabriel's theorem.
- To be able to explain the sketch of the proof of Gabriel's theorem.
Keywords
Representations of quivers, Path algebras, Weyl groups, Root systems, Gabriel's theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course. Assignments will be given during class sessions.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Quivers and their representations | Details will be provided during each class session. |
| Class 2 | Path algebras, Krull--Schmidt theorem | Details will be provided during each class session. |
| Class 3 | Euler forms, Tits forms | Details will be provided during each class session. |
| Class 4 | Representation spaces | Details will be provided during each class session. |
| Class 5 | Reflection functors | Details will be provided during each class session. |
| Class 6 | Weyl groups, Root systems | Details will be provided during each class session. |
| Class 7 | Gabriel's theorem | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None in particular.
Reference books, course materials, etc.
・I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006. x+458 pp.
・A. Kirillov Jr., Quiver Representations and Quiver Varieties, Grad. Stud. Math., 174, American Mathematical Society, Providence, RI, 2016. xii+295 pp.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A507 ： Advanced topics in Algebra G1
Prerequisites
Basic knowledge on algebra is expected.
Other
None in particular.