Hadi Salmasian: Lie Groups and Quantization

4th September, 2024

This course is currently ongoing, so not all links will be active.

This is a 23-lecture course, with each lecture being around 80 minutes long, given by Hadi Salmasian.

The goal of the course is to first cover the foundational theory of Lie groups and then move on to more advanced topics that expose the audience to areas of active research. The following is the list of topics that are intended to be covered:

  • Foundational theory of Lie groups: Lie groups, the exponential map, Lie correspondence. Homomorphisms and coverings. Closed subgroups. Classical groups: Cartan subgroups, fundamental groups. Manifolds. Homogeneous spaces. General Lie groups.
  • Introduction to quantization: Symplectic manifolds, pre-quantization, the orbit method. Poisson manifolds, Manin triples. Universal enveloping algebras, quantum sl(2) and its representations, quantum symmetric spaces.

Prerequisites: Strong background in 2nd/3rd year level algebra and real analysis is required. In particular, students should be familiar with basic group theory (e.g., normal subgroups, quotients, Lagrange’s theorem, isomorphism theorems, characterization of finite abelian groups) and elementary analysis (e.g., metric spaces, compactness, Heine-Borel Theorem, uniform convergence, series of functions).

Textbooks: The lectures on the foundational theory of Lie groups will be based on the following textbook:

  • Hall, B., Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. 2nd Edition, Springer, 2015.

The more advanced material on quantization will be covered from various references, including the following reference:

  • Chari, V., Pressley A., A Guide to Quantum Groups, Cambridge, 1995.
  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
  9. Lecture 9
  10. Lecture 10
  11. Lecture 11
  12. Lecture 12
  13. Lecture 13
  14. Lecture 14
  15. Lecture 15
  16. Lecture 16
  17. Lecture 17
  18. Lecture 18
  19. Lecture 19
  20. Lecture 20
  21. Lecture 21
  22. Lecture 22

These videos were produced by the Fields Institute, as a graduate course (link to course page).