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Quantum groups and knot theory

(Fall 2007)

The master course Quantum groups and knot theory (course description) is part of the national Dutch master program, as well as of the master class 2007-2008 Quantum groups, affine Lie algebras and their applications.

Schedule: Wednesday, 10:15-13:00.
Location: Buys Ballot Laboratory, room 105b, De Uithof, Princetonlaan, Utrecht.
As of November 14, the lectures will be given in Room 611AB, Utrecht Mathematical building, Budapestlaan 6.

In the first three weeks (week 37, 38 and 39) an introduction to knot theory is given by Arjeh Cohen, assisted by Dan Roozemond.
In the second part an introduction to the theory on quantum invariants of ribbon links and knots is given by Eric Opdam and Jasper Stokman. Detailed information on the second part of the course will be given on this homepage.

Literature: In the second part of the course we will use the following book
[1] C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, Panoramas et Syntheses, no. 5 (1997), Societe Mathematique de France, ISBN 2-85629-055-8.

For each week we provide a text (pdf) with additional remarks and exercises. It also contains the homework exercises, which should be handed in one week later. The additional texts can be downloaded below, where the content per week is described.

Other useful references are
[2] C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer Verlag.
[3] V.G. Turaev, Quantum invariants of knots and 3-manifolds, W. de Gruyter, Berlin, 1994.

Program

 Weeks 37-39
See the homepage of Arjeh Cohen's mini-course on knot theory.

Weeks 40-42 and 44-51
October 3: Yang-Baxter equation, braid groups and Hopf algebras (References: [1], Chapter 1 and part of chapter 2 and the syllabus (version December 3) with additional comments and exercises).
Homework: The homework exercises are exercises (b), (e) (i) and (ii), (h), (k) (iii) and (iv), and exercise (l) in the syllabus.
October 10: Hopf algebras and braided monoidal categories (Reference: [1], Chapter 2 and the syllabus (version of December 3) with additional comments and exercises).
Homework: The homework exercises are exercises (c), (f) and (i) in the syllabus.
Note: The explicit expression of the universal R-matrix in the homework exercise (i) in the syllabus (which is exercise 4.4(c) in chapter 2 of [1]) contains a mistake. It suffices for the homework to give the solution of the exercise when lambda=0 (in which case the universal R-matrix is correct).
The correct expression for the universal R-matrix for arbitrary lambda is given in [2], Section VIII.2, Example 2.
October 17: The quantum double (Reference: [1], Chapter 3 and the syllabus).
Homework: Exercises 2.4, 2.9(ii), 3.2 and 3.6 in the syllabus.
October 24: No lecture (autumn break).
October 31: Quantum double and quantum sl(2) (Reference: the syllabus, final version put on the web-site on November 1).
Homework: Exercises 1.4, 1.9, 2.3 and 2.11 of the syllabus.
November 7: Universal R-matrix of quantum sl(2). Hecke algebras,Temperley-Lieb algebra, and quantum sl(n). (Reference: the syllabus).
Homework: Exercises 2.3, 2.7 and 3.9 of the syllabus.
November 14: Skein categories (Reference: [1], Chapter 5 and the syllabus).
Homework: Exercises (a), (b), (e) and (i) of the syllabus.
November 21: Representation theory of quantum sl(2) and of the skein category (Reference: [1], Chapter 7 and the syllabus).
Homework: Exercises 1.7, 2.2 and 3.4 of the sylabus.
November 28: Ribbon categories and Reshetikhin-Turaev invariants (Reference: [1], Chapter 6 and the syllabus (version of December 3)).
Homework: Exercises (b), (f), (i) (1-4) of the syllabus.
Two addition comments on exercise (i):
If x,y in G are objects of V we define their tensor product as their product xy in G.
If f,g in K are morphisms in V we define their tensor product as their product fg in K.
December 5: Ribbon algebras and ribbon categories (Reference: [1], Chapter 6 and the syllabus (version of December 4)).
Homework: Exercise (i) (5-7) in the syllabus of week 11 and Exercise (c) (1-3) in the syllabus of week 12.
Requist: Please download the evaluation form for the course, fill it in, and hand it in at the lectures on Wednesday. Thanks in advance!
December 12: Reshetikhin-Turaev invariants and the colored Jones polynomial (Reference: [1], Chapter 7 and the syllabus (version of December 12)).
There is no homework.

Last modified: Wednesday, 12-Dec-2007 15:34:32 CET