Quantum Groups

This class meets Tuesdays and Thursdays 1:30 - 2:50 PM in room 20-21G. This is the Human Biology building in the inner courtyard of the main quadrangle.

Office Hours Tuesdays and Thursdays 12:15-1:15 in 383U.

Canvas page for this class.

Lecture Notes

• Lecture of Sept. 27, 2022. Hopf algebras and monoidal categories.
• Lecture of Sept. 29, 2022. Braided categories. Sweedler notation.
• Lecture of Oct 4, 2022. Rigid categories and duality.
• Lecture of Oct 6, 2022. Quasitriangular Hopf algebras.
• Lecture of Oct 10, 2022. Examples of Quasitriangular Hopf algebras. $U_q(\mathfrak{sl}_2)$. Dual pairings of Hopf algebras
• Lecture of Oct 13, 2022. Ribbon Categories. More about QTHA.
• Lecture of Oct 18, 2022. Solvable Lattice Models.
• Lecture of Oct 20, 2022. The morphism $\mathfrak{u}:V\to V^{\ast\ast}$, and the corresponding element of $H$. A look at Turaev and Reshetikhin.
• Lecture of Oct 25, 2022. More about $\mathfrak{u}$. Convolution Theory. Drinfeld Double (I).
• Lecture of Oct 27, 2022. Ribbon Hopf algebras, Ribbon Categories, Drinfeld Double (continued)
• Lecture of Nov 1, 2022. Drinfeld Double (continued), Kauffman bracket
• Lecture of Nov 10, 2022. The quadratic relation for the R-matrix in $U_q(\mathfrak{gl}(n))$: Schur-Weyl-Jimbo duality and the skein relation for the Jones polynomial
• Lecture of Nov 15, 2022. The quadratic relation for the R-matrix in $U_q(\mathfrak{gl}(n))$: Schur-Weyl-Jimbo duality and the skein relation for the Jones polynomial (continued)
• Lecture of Nov 17. 2022. Drinfeld Double (wrap-up). Modular Tensor Categories.
• Lecture of Nov 29. 2022. Modular Tensor Categories (continued).
• Lecture of Dec 1. 2022. Modular Tensor Categories (continued).
• Lecture of Dec 6. 2022. Quantum Groups at roots of unity. The Lusztig quantum group, Tilting modules and the MTC $\mathcal{C}(\mathfrak{g},k)$.

Older Lecture Notes

Here are lecture notes from a previous course I taught on the same subject.

• 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.