# Quantum Groups

This class meets 10:30-11:45 Mondays and Wednesdays in 384I.

- Lecture 1. Introduction. Bialgebras. Hopf Algebras. Examples of Hopf Algebras.
- Lecture 2. Sweedler notation. Bialgebras and monoidal categories. Rigid categories. Braided monoidal categories.
- Lecture 3. Braided categories. Quasitriangular Hopf algebras. Quantized enveloping algebras. Reidemeister moves. Braiding in a rigid category.
- Lecture 4. Tangles and Framed Tangles. The naive trace. Reflexivity and twisting. The ribbon trace. Ribbon categories.
- Lecture 5. Review of Tangles and Ribbon Categories. $u_V$, $v_V$, $\theta_V$ and all that. Quasitriangular Hopf algebras. The element $\mathbf{u}$.
- Lecture 6. The standard module of $U_q(\mathfrak{sl}_2)$ and its R-matrix. The $6$-vertex model.
- Lecture 7. Ribbon Hopf Algebras. Schur-Weyl-Jimbo duality.
- Lecture 8. Kauffman brackets and the Jones polynomial.
- Lecture 9. Duality in Ribbon Categories. The Kauffman bracket as a quantum trace. Gaussian binomial coefficients
- Lecture 10. Borel subgroups. New Hopf Algebras from Old. A duality property of QTHA. Quantized enveloping algebras
- Lecture 11. Convolution. The Drinfeld Double (I). The RTT equation.
- Lecture 12. The Drinfeld Double (II). Modular Tensor Categories.
- Lecture 13. Review: $U_q(\mathfrak{sl}_n)$. The Drinfeld Double (III). Modular Tensor Categories (II).
- Lecture 14. Modular Tensor Categories (III). The $SL(2,\mathbb{Z})$ action.
- Lecture 15. The Racah-Speiser algorithm. The Kac-Walton formula.
- Lecture 16. Lusztig's quantum group. Tilting modules. The Fusion Category in Action.
- Lecture 17. Affine Lie algebras. The fusion ring.
- Lecture 18. Kashiwara Crystals. Crystal bases and quantum groups. Crystals of tableaux. A peek at tableau combinatorics.

## Exercises

There are exercises at the last page in Lectures 2,3,4,5 and 9.

## Recommended texts

The following texts should be helpful. Majid's text is wonderful
for getting to the heart of things very quickly, and Kassel's
longer book also contains a lot of useful material.

- Majid, A primer of quantum groups.
- Kassel, Quantum groups

The course also used:

- Turaev, Quantum Invariants of Knots and 3-Manifolds
- Bakalov and Kirillov, Lectures on tensor categories and modular functors

and various papers cited in the lectures.