Advanced topics in Algebra C

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.A403

Number of credits

100

Course offered

2026

Offered quarter

3Q

Syllabus updated

Mar 5, 2026

Language

English

Syllabus

Course overview and goals

This is an introductory course on the representation theory of quantum groups. The quantum groups in this course are the Drinfeld–Jimbo quantum groups (quantized enveloping algebras), which are Hopf algebras that can be regarded as a "q-deformation" of the universal enveloping algebras of semisimple Lie algebras. The representation theory of quantum groups can be viewed as a deformation of the representation theory of semisimple Lie algebras by a parameter q. It provides many important topics, including the construction of nontrivial solutions to the quantum Yang–Baxter equation, the theory of crystal bases and canonical bases, and analogies with modular representation theory through the specialization of q to the roots of unity.

In this course, we will start from the definition of quantum groups, and learn the basics of their representation theory. We also provide an overview of the theory of crystal bases and canonical bases. The aim of this course is to understand how the parameter q deforms the theory of semisimple Lie algebras and to study the applications of such deformations.

This course is followed by “Advanced Topics in Algebra D”.

Course description and aims

- To be able to explain the definition of quantized enveloping algebras associated with root data.
- To be able to explain the classification of integrable highest weight modules over quantized enveloping algebras.
- To be able to explain the construction of universal R-matrix.
- To be able to explain the properties of crystal bases and canonical bases.

Keywords

Quantized enveloping algebras, Integrable highest weight modules, Universal R-matrix, Crystal bases, Canonical bases

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

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.
・J.C. Jantzen, Lectures on quantum groups, Grad. Stud. Math., 6, American Mathematical Society, Providence, RI, 1996.
・J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Grad. Stud. Math., 42, American Mathematical Society, Providence, RI, 2002.

Evaluation methods and criteria

Assignments (100%)

Related courses

• MTH.A404 ： Advanced topics in Algebra D

Prerequisites

It is desirable to have basic knowledge of algebra.

Other

None in particular.