Special lectures on advanced topics in Mathematics B

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Hironori Oya / Satoshi Naito

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (本館2階201数学系セミナー室)

Class

-

Course Code

MTH.E432

Number of credits

200

Course offered

2022

Offered quarter

3Q

Syllabus updated

Jul 10, 2025

Language

Japanese

Syllabus

Course overview and goals

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.

Course description and aims

・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

• 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/Objectives

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 course material.

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

Evaluation methods and criteria

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

Basic knowledge on algebra is expected.