MAGIC109: Introduction to Hopf algebras and quantum groups

Course details

A core MAGIC course

Semester

Spring 2024
Monday, January 29th to Friday, March 22nd; Monday, April 22nd to Friday, May 3rd

Hours

Live lecture hours
10
Recorded lecture hours
10
Total advised study hours
80

Timetable

Wednesdays
13:05 - 13:55 (UK)

Description

Quantum groups are a manifestation of symmetry in its most abstract algebraic form. One way to motivate quantum groups is to observe that a group acting on a manifold M also acts on its algebra of functions, F(M). One can then substitute F(M) with a more general, noncommutative algebra, which leads to the notion of a Hopf algebra as something capable of acting on such a `noncommutative space'. The theory was reinvigorated by examples due to Drinfeld and Jimbo in the 1980s, inspired by quantum mechanics; the term `quantum group' was coined. Hopf algebras and their representations found applications in many fields including topology, number theory, mathematical physics, quantum information theory and, recently, finance (via stochastic calculus). 

In the course we adopt a hands-on approach to Hopf algebras: we build on accessible examples arising from groups and Lie algebras, and learn to present new algebras by generators and relations. We develop tensor calculus and emphasise the idea of duality between algebras and coalgebras, modules and comodules etc to approach the celebrated quantum group constructions due to Drinfeld-Jimbo and Manin. Self-duality of these objects is expressed by an 'R-matrix', or quasitriangular structure as popularised by Drinfeld and Majid. To give an example of an application of quantum groups, we look at a quasitriangular structure for the quantum group Uq(sl2) to see how it gives rise to knot invariants.

It should be noted that our approach is purely algebraic; the course aims to equip the students with a suitable background to further explore analytic, number-theoretic, geometric, topological and physical aspects of Hopf algebras.

Prerequisites

  • Essential: undergraduate linear algebra, group theory, ring theory. 
  • Advantageous: representation theory, Lie algebras. 
All the necessary definitions will be introduced in the course, but students should be prepared to adapt to new notation and new ways of looking at familiar algebraic concepts. 

Syllabus

Topics marked "if time" may be left as optional reading assignments; in this case, they will not be examinable.
  • Linear and multilinear algebra: tensor products, dual spaces, quotients.
  • Presentation of algebras using generators and relations. Symmetric algebras, universal enveloping algebras.
  • Coalgebras and their representations. (If time: the fundamental theorem on coalgebras.) 
  • Bialgebras and Hopf algebras. Sweedler notation. Examples, e.g. group algebras.
  • Application: proof of the Poincare-Birkhoff-Witt theorem using Hopf algebra properties. 
  • (Co)actions of Hopf algebras on algebras. Quantum symmetries. 
  • Duality pairing. (If time: the Drinfeld double; the Heisenberg algebra.)
  • The Drinfeld-Jimbo quantum group Uq(sl2). q-calculus and q-deformations. 
  • Quasitriangular structures. The quantum Yang-Baxter equation. Braidings. (If time: self-duality and the discrete Fourier transform.)
  • Application: invariants of knots and links arising from representations of a quantum group.

Lecturer

  • Dr Yuri Bazlov

    Dr Yuri Bazlov

    University
    University of Manchester

Bibliography

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Assessment

The assessment for this course will be released on Monday 13th May 2024 at 00:00 and is due in before Friday 24th May 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

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Lectures

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