Representation Theory and Knot Invariants

Summer semester 2019

Dates:
Lectures Monday 9.45-11.15 in 7.527 and Friday 9.45-11.15 in 7.527
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.

Contents:
Starting with a short and informal introduction to knot invariants, we will try to understand the algebraic background of constructions of knot polynomials, for instance the Jones polynomial (Fields medal awarded to VFR Jones, 1990). This includes topics such as braid groups, quantum groups and R-matrices, Hopf algebras, Temperley-Lieb algebras and other diagram algebras, etc. The main objects to study are algebras, representations and tensor products of representations. Thus, the course provides many examples for representation theory. The course can be a basis for master theses and for subsequent seminars on various levels, depending on the interests of the participants.

Prerequisites:
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.

Contents of lectures:
Chapter one: Knots, links and polynomials.
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.

Chapter two: Braids.
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.

Chapter three: Knots and braids.
Monday, April 29: Closing braids. Markov moves, Markov functions. Representations. Tensor products.
Thursday, May 2: Properties of tensor products.

Chapter four: The Burau representation.
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.

Chapter five: Hecke algebras.
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 Hn and Hn+1. Ocneanu trace.
Thursday, May 16: The HOMFLY-PT polynomial.

Chapter six: Quantum Yang-Baxter equation and quasi-triangular Hopf algebras.
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.

Chapter seven: Lie algebras, enveloping algebras and quantised enveloping algebras.
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.

Chapter eight: The FRT construction.
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.

Chapter nine: Polynomials and matrices.
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 sheets:
Problem sheet 1
Problem sheet 2
Problem sheet 3
Problem sheet 4
Problem sheet 5

References:
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).