PhD Course
Introduction to Quantum groups
and Quantum Loop Algebras

Lecture notes

The aim of this course is to provide a brief introduction to the theory of quantum groups.

The first part of the course will discuss the parallel between the classical highest-weight integrable representation theory of the Lie algebra sl(2) and that of its quantum counterpart Uqsl(2) (and its generalization to Kac-Moody algebras), with a special focus on the role of the universal R-matrix and the Yang-Baxter equation. The second part of the course will focus on quantum loop algebras - that is, quantum groups associated to affine Lie algebras - and their finite-dimensional representations.

  • Lecture 1. Irreducible representations of sl(2) and quantum sl(2). [Notes] [Video]
  • Lecture 2. Quasitriangular Hopf algebras and the R-matrix of quantum sl(2). [Notes] [Video]
  • Lecture 3. The quantum double construction. [Notes] [Video]
  • Lecture 4. Kac-Moody algebras and quantum groups. [Notes] [Video]
  • Lecture 5. Representations of quantum groups. [Notes] [Video]
  • Lecture 6. Weyl groups. [Notes] [Video]
  • Lecture 7. Quantum Weyl groups. [Notes] [Video]
  • Lecture 8. Affine Lie algebras and affine Weyl groups. [Notes] [Video]
  • Lecture 9. Quantum loop algebras. [Notes] [Video]
  • Lecture 10. Rationality of fields. [Notes] [Video]
  • Lecture 11. Classification of finite-dimensional irreducible representations. [Notes] [Video]
  • Lecture 12. Evaluation representations of quantum affine sl(2). [Notes]

Information (dated to 2022)

Every Monday and Tuesday

@ 15:00-17:00 CET

Jan. 10 - Feb. 15, 2022

If you are interested in joining the course, please email me at andrea.appel@unipr.it.

The course will be held entirely online with lectures on Monday/Tuesday from 3pm to 5pm. The course will start on Monday, January 10th, and will end on Tuesday, February 15th. Lectures will be streamed on Teams at the following link

MS Teams

Recording of the lectures will be available online (with restricted access).