Quantum Groups and Integrable Systems

Course manual 2017/2018

Course content

Typical examples of quantum groups are noncommutative deformations of the algebra of functions on a Lie group, or deformations of the universal enveloping algebra of a Lie algebra. We consider an important subclass of Hopf algebras with particularly rich additional algebraic structure (a nontrivial universal braiding) which turns out to have powerful applications in knot theory and integrable systems. It is the aim of this course to explain the basic structures of quantum groups, both from an algebraic point of view and from a categorical point of view, and to discuss some of their applications in knot theory and integrable systems.
Topics that will be treated are: braided Hopf algebras and braided tensor categories, including the examples arising from quantised universal enveloping algebras and Drinfeld's quantum double construction. Application to knot theory, including skein theory and the Jones polynomial. Application to integrable systems, including the Heisenberg spin chain and the algebraic Bethe ansatz method.

Prerequisites: Basic knowledge of algebra and representation theory (of finite groups). Some knowledge of Lie groups and Lie algebras will be very helpful, but it is not necessary.

Study materials

Literature

• 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

Objectives

• Is acquainted with the concept of quantum groups and the relation to Lie algebras, braided Hopf algebras and braided tensor categories.
• Understands how topological invariants can be constructed using quantum groups.
• Has learned the basics of the algebraic Bethe ansatz method for integrable models from statistical physics.
• Knows how Heisenberg models from statistical physics can be constructed and analysed using quantum groups.

Teaching methods

• Lecture

Learning activities

Attendance

The programme does not have requirements concerning attendance (OER-B).

Assessment

Inspection of assessed work

Contact the course coordinator to make an appointment for inspection.

Assignments

Homework exercises

• Biweekly homework exercises

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.uva.nl/plagiarism

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• prof. dr. Jasper Stokman