Problem classes Thursday 9.45-11.15 in 7.527

Lectures start on Monday, 8th of April. In the first two weeks, the Thursday slot will be used for lectures.

This is a master course that also can be taken as an advanced bachelor course.

Prerequisites: Linear Algebra I and II, Algebra I, familiarity with and interest in abstract mathematical structures. Parallely attending Algebra II makes sense, and an introductory course on general representation theory (to be offered in the winter semester 2019/20) can be used to get a broader picture of representation theory and as a starting point for further directions.

Monday, April 8: Knots and links, topological and combinatorial equivalences. Knot diagrams, Reidemeister moves.

Thursday, April 11: Decidability, computability, complexity. Invariants.

Friday, April 12: Kauffman bracket, Tait number (writhe) and Jones polynomial. Example: Hopf link.

Monday, April 15: Geometric braids and braid diagrams. Group structure, generators. Free groups.

Thursday, April 18: Construction of free groups. Presentations. Artin braid group, braid relations.

Friday, April 19: Good Friday.

Monday, April 22: Easter Monday.

Thursday, April 25: Problem class.

Friday, April 26: Pure braids.

Monday, April 29: Closing braids. Markov moves, Markov functions. Representations. Tensor products.

Thursday, May 2: Properties of tensor products.

Thursday, May 2: Burau representation and reduced Burau representation. Subrepresentations and quotient representations.

Friday, May 3: From the reduced Burau representation to a Markov function and the Alexander-Conway polynomial.

Friday, May 3: Modules. Algebras.

Monday, May 6: The Iwahori-Hecke algebra. Statement of main properties. Some facts on symmetric groups. Lemmas.

Thursday, May 9: Problem class.

Friday, May 10: More lemmas. Freeness and basis. Exchange condition and consequences.

Monday, May 13: Comparing H

Thursday, May 16: The HOMFLY-PT polynomial.

Thursday, May 16: Motivation. QYBE and braid relations.

Friday, May 17: Examples. Coalgebras and algebras.

Monday, May 20: Examples. Bialgebras. Tensor algebra.

Thursday, May 23: Problem class.

Friday, May 24: Tensor algebra, continued. Hopf algebras.

Monday, May 27: Problem class.

Friday, May 31: No lecture.

Monday, June 3: Examples. Tensor products of modules and homomorphism spaces. Quasi-cocommutative, universal R-matrix, quasi-triangular (braided).

Thursday, June 6: Properties of the universal R-matrix of a quasi-triangular bialgebra. Maps between a Hopf algebra and its dual.

Friday, June 7: From the universal R-matrix to solutions of QYBE.

Friday, June 7: Lie algebras, homomorphisms, representations. Examples. Universal enveloping algebras.

Pentecost holidays.

Monday, June 17: PBW theorem, filtered and graded algebras. Comultiplication.

Thursday, June 20: Corpus Christi.

Friday, June 21: Finite dimensional representations of sl(2) over the complex numbers. Solvable, simple and semisimple Lie algebras. Jacobson-Morozov theorem.

Monday, June 24: Problem class.

Thursday, June 27: The quantised universal enveloping algebra of sl(2).

Friday, June 28: Finite dimensional representations when q is not a root of unity. Harish-Chandra homomorphism.

Monday, July 1: Verma modules. Semisimplicity.

Thursday, July 4: Semisimplicity, continued. The case of q being a root of unity.

Friday, July 5: More on the case of q being a root of unity. Back to the case of q not being a root of unity: tensor products.

Monday, July 8: Solutions of QYBE from finite dimensional representations.

Monday, July 8: Cobraided bialgebras.

Thursday, July 11: Solutions of QYBE from universal r-forms. Beginning of the proof of the FRT theorem: Construction of the bialgebra A(c).

Friday, July 12: Proof, continued.

Monday, July 15: Example.

Monday, July 15: Polynomials and varieties. Monoids, groups and duals. The bialgebras M(2), GL(2) and SL(2). Quantisations.

Thursday, July 18:

Friday, July 19: Problem class

Problem sheet 1

Problem sheet 2

Problem sheet 3

Problem sheet 4

Problem sheet 5

Adams, The knot book

Burde and Zieschang, Knots

Cromwell, Knots and links

Kauffman, Knots and physics

Murasugi, Knot theory and its applications

Sossinski, Knots

Chari and Pressley, A guide to quantum groups

Humphreys, Reflection groups and Coxeter groups

Jantzen, Lectures on quantum groups

Kassel, Quantum groups

Kassel, Rosso and Turaev, Quantum groups and knot invariants

Kassel and Turaev, Braid groups

Erdmann and Wildon, Introduction to Lie algebras.

Humphreys, Introduction to Lie algebras and representation theory.

(more references to be added)

Pictures of knots, history, ... (University of Wales, Bangor).