2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra D
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hironori Oya
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.A404
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 4Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course follows “Advanced Topics in Algebra C” and provides an exposition of the representation theory of quantum groups together with related advanced topics. The aim of this course is to learn the theory of crystal bases and canonical bases, as well as a quantization of spaces via the representation theory of quantum groups.
Course description and aims
- To be able to explain the relations between quantized enveloping algebras and modified quantized enveloping algebras.
- To be able to explain the properties of canonical bases of modified quantized enveloping algebras.
- To be able to explain the definition of quantized coordinate rings.
Keywords
Modified quantized enveloping algebras, Extremal weight modules, Based modules, Quantized coordinate rings
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 the classes.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Modified quantized enveloping algebras |
Details will be provided during each class session |
| Class 2 | Extremal weight modules |
Details will be provided during each class session |
| Class 3 | Based modules |
Details will be provided during each class session |
| Class 4 | Quantized coordinate rings (1) |
Details will be provided during each class session |
| Class 5 | Quantized coordinate rings (2) |
Details will be provided during each class session |
| Class 6 | Advanced topics (1) |
Details will be provided during each class session |
| Class 7 | Advanced topics (2) |
Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to explore references provided in the lectures.
Textbook(s)
None required.
Reference books, course materials, etc.
・G. Lusztig, Introduction to quantum groups, Reprint of the 1994 edition. Mod. Birkhäuser Class. Birkhäuser/Springer, New York, 2010.
・T. Shoji, Chevalley groups and algebraic groups, Nippon Hyoron sha co.,Ltd., 2025.
Evaluation methods and criteria
Assignments (100%)
Related courses
- MTH.A403 ： Advanced topics in Algebra C
Prerequisites
It is desirable to have basic knowledge of algebra.
Other
None in particular.