Veranstalter:

Ralf Holtkamp and Louis-Hadrien Robert

Time and place:

On Thursday at 10:30 in Room 1240 (Geomatikum).

Description:

In this seminar we will define and study quantum invariants of knots and 3-manifolds. We will begin by a few talks about semi-simple Lie algebras and their representations. The planned talks are:

• 1 Introduction and overview. 02.04.2015 (LHR)

• 2 Lie Algebras, semi-simplicity, Killing's form. Following [H, §1, § 2.(4-6)]. 09.04.2015 (SM)

• 3 Representation of sl(2,C), roots, root systems. Following [H, § 2.(7-8), §3.9]. 16.04.2015 (LW)

• 4 Representations of sl(n, C) Following. [FH, §15].

• 5 Definition of the Uq(sl(n,C)) and first properties. Following [KRT, §:4].

• 6-7 Invariant of tangles, links and 3-manifolds. Following [KRT, §5-6] and [O, §8].

• 8 Representation of Uq(sl(n),C). Following [KRT, §7].

• 9 Vassiliev invariants. Following [KRT, §8]. June 2015 (EE)

• 10/11 Categorification of the Jones polynomial. Following [BN].

Literature:

 [BN] D. Bar-Natan, On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology, MSP.

 [FH] W. Fulton and J. Harris, Representation Theory. A first course, Springer.

 [KRT] C. Kassel, M. Rosso and V. Tuarev, Quantum groups and knot invariants, SMF.

 [H] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer

 [O] T. Ohtsuki, Quantum Invariants, World Scientific.

Prerequisites:

The seminar is aimed at Master students in mathematics or mathematical physics, but is also open for Bachelor students who are familiar with the contents of an Algebra I course.

Remarks:

Depending on the interest of the participants, we may treat the following additional subjects:

• Rasmussen invariant,
• Kirby moves,
• Homfly-pt polynomial.

First Meeting:

A first meeting took place on Thursday 05/02/2015. If you are interested, but you did not attend this meeting, please send an email to firstname.lastname@uni-hamburg.de (replace the names by some adequate values).