# 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).

## 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

Also relevant:

• Turaev, Quantum Invariants of Knots and 3-Manifolds

and the two 1990 papers of Turaev and Reshetikhin.

For the Drinfeld double, there are discussions in Majid and Kassel, but we recommend Reshetikhin and Semenov-Tian-Shansky, Quantum R-matrices and factorization problems (1988).