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.
・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 ring.
・Understand applications of Fomin-Zelevinsky's cluster algebras to the representation theory of quantum affine algebras.
quantum affine algebra, q-character, quantum Grothendieck ring, Kazhdan-Lusztig algorithm, cluster algebra
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
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. |
None in particular.
T. Nakanishi: Cluster Algebras and Scattering Diagrams, Part I: Basics in Cluster Algebras; arXiv:2201.11371
Assignments (100%)
Basic knowledge on algebra is expected.