Math 607: SPRING 2025, TR 14:00-15:20, 373 McKenzie

      QUANTUM GROUPS
Arkady Berenstein

The course is about algebraic aspects of quantum groups. Quantum groups (e.g., quantized enveloping algebras) were introduced independently by Drinfeld and Jimbo around 1985, as an algebraic framework for quantum Yang-Baxter equations. Since then numerous applications of quantum groups have been found in areas ranging from theoretical physics via symplectic geometry and knot theory to ordinary and modular representations of reductive algebraic groups. The course provides a systematic introduction to the structure theory and representation theory of quantum groups.

I am planning to cover the following topics:

• Groups, Bialgebras, and Hopf algebras
• Modules, comodules, and bimodules
• Quantum linear algebra (after Manin)
• Braided Linear Algebra (after Majid)
• Tensor and Braided categories
• (Re)construction of Hopf algebras
• Quantum algebraic groups
• Quantized enveloping algebras and their representations

I then plan to emphasize the following additional topic if time permits:

• Drinfeld's theory of quantum deformations of  Poisson-Lie groups and Lie bialgebras. The main result is the theorem by Kazhdan and Etingof, 1995, which states that the universal enveloping algebra of every Lie bialgebra admits a canonical deformation-quantization.
• Kontsevich deformation of Poisson brackets on commutative algebras. The main result is the theorem by Kontsevich, 1997, which states that the coordinate algebra of any Poisson variety admits a canonical deformation-quantization

REFERENCE BOOKS.

[1] S. Majid, Foundations of Quantum Group theory.
[2] S. Montgomery, Hopf Algebras and Their Actions on Rings.
[3] Y. Manin, Quantum groups and non-commutative geometry.
[4] G. Lusztig, Introduction to Quantum Groups.
[5]  J. Jantzen, Lectures on Quantum Groups.
[6] A. Joseph, Quantum groups and their primitive ideals.
[7]  Brown and Goodearl, Lectures on algebraic quantum groups.

Prerequisites: 600 Algebra.  Some knowledge of Category Theory and Representation Theory of Lie algebras & groups would also be helpful.