Mikhail Bershtein

Quantum Groups. 2027 (SISSA)

Dates: February–April 2027.

Quantum groups were introduced in the mid-1980s and very quickly became one of the most important topics in mathematics and mathematical physics. They are still actively studied, and knowledge of them is necessary for work in many areas.

The purpose of this course is to provide an introduction to quantum groups. The content will be based on classic works of the 1980s and early 1990s; we will not cover the latest results. No prior knowledge of quantum groups is assumed, but familiarity with Lie algebras and groups, Poisson brackets, and the basic notions of category theory is expected.

Plan of the Course (Tentative)

  1. Poisson algebras and quantization. Poisson-Lie groups and Lie bialgebras. Classical r-matrices. Dual Poisson-Lie groups.
  2. Quantum groups and algebras. Hopf algebras. Quantum R-matrices.
  3. Drinfeld-Jimbo quantum groups. Finite-dimensional representations of Uq(g).
  4. Drinfeld double. Drinfeld's theorem.
  5. RTT realization. Functions on quantum SL(n).
  6. Lusztig's group. Factorization of the universal R-matrix.

References

  1. V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.
  2. P. Etingof and O. Schiffmann, Lectures on Quantum Groups, second edition, International Press, 2002.
  3. J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, 1996.