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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Oya Hironori Naito Satoshi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (本館2階201数学系セミナー室)
- Group
- -
- Course number
- MTH.E432
- Credits
- 2
- Academic year
- 2022
- Offered quarter
- 3Q
- Syllabus updated
- 2022/4/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The main topic of this course is the finite dimensional representation theory of quantum affine algebras. Quantum affine algebras are Hopf algebras, which are regarded as q-analogues of the universal enveloping algebras of certain infinite dimensional Lie algebras, called affine Lie algebras. The finite dimensional representations of such algebras have been extensively studied since the mid-1980s, but the investigation of them is still active and some fundamental problems remain unsolved. In this course, we focus on the problem of the calculation of the q-characters of irreducible representations, and explain the fundamental known facts together with recent results.
The first half of the course is devoted to the explanation of the well-known topics in the finite dimensional representation theory of quantum affine algebras, including highest weight theory, q-characters, the calculation of irreducible q-characters via Kazhdan-Lusztig algorithm. I would like to explain the arguments frequently appearing in representation theory through this part. In the second half of the course, I talk about some recent topics, including applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras and similarities in the representation theory of quantum affine algebras of several different Dynkin types. If time permits, I will explain the representation theory of certain variants of quantum affine algebras, for example, shifted quantum affine algebras and the Borel subalgebras of quantum affine algebras. Through this part, I would like to show examples of research themes in this field.
Student learning outcomes
・Understand the classification of finite dimensional irreducible representations of quantum affine algebras.
・Understand the definition of the q-characters of finite dimensional representations of quantum affine algebras.
・Understand several constructions of the quantum Grothendieck ring of the monoidal category of finite dimensional representations of a quantum affine algebra, and understand the procedure of the Kazhdan-Lusztig algorithm in the quantum Grothendieck rings.
・Understand applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras.
Keywords
quantum affine algebra, q-character, quantum Grothendieck ring, Kazhdan-Lusztig algorithm, cluster algebra
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Classification of finite dimensional irreducible representations of quantum affine algebras 1 | Details will be provided during each class session. |
| Class 2 | Classification of finite dimensional irreducible representations of quantum affine algebras 2 | Details will be provided during each class session. |
| Class 3 | q-characters of finite dimensional representations of quantum affine algebras 1 | Details will be provided during each class session. |
| Class 4 | q-characters of finite dimensional representations of quantum affine algebras 2 | Details will be provided during each class session. |
| Class 5 | Several constructions of the quantum Grothendieck ring of the monoidal category of finite dimensional representations of a quantum affine algebra 1 | Details will be provided during each class session. |
| Class 6 | Several constructions of the quantum Grothendieck ring of the monoidal category of finite dimensional representations of a quantum affine algebra 2 | Details will be provided during each class session. |
| Class 7 | Kazhdan-Lusztig algorithm for the quantum Grothendieck ring 1 | Details will be provided during each class session. |
| Class 8 | Kazhdan-Lusztig algorithm for the quantum Grothendieck ring 2 | Details will be provided during each class session. |
| Class 9 | Applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras 1 | Details will be provided during each class session. |
| Class 10 | Applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras 2 | Details will be provided during each class session. |
| Class 11 | Applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras 3 | Details will be provided during each class session. |
| Class 12 | Similarities in the representation theory of quantum affine algebras of several different Dynkin types 1 | Details will be provided during each class session. |
| Class 13 | Similarities in the representation theory of quantum affine algebras of several different Dynkin types 2 | Details will be provided during each class session. |
| Class 14 | Representation theory of certain variants of quantum affine algebras 1 | Details will be provided during each class session. |
| Class 15 | Representation theory of certain variants of quantum affine algebras 2 | Details will be provided during each class session. |
Textbook(s)
None in particular.
Reference books, course materials, etc.
T. Nakanishi: Cluster Algebras and Scattering Diagrams, Part 1: Basics in Cluster Algebras; arXiv:2201.11371
Assessment criteria and methods
Assignments (100%)
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic knowledge on algebra is expected.
