MAGIC109: Introduction to Hopf algebras and quantum groups

Course details

Semester

Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th

Hours

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

Timetable

Wednesdays
10:05 - 10:55

Course forum

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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, mathematical physics and, recently, quantum information theory. 


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.

We look at quasitriangular structures in simple cases (finite groups, polynomial algebras) and discuss their links to cohomology of groups, quantum mechanics and representation theory.

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, 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

  • Linear and multilinear algebra: tensor products, dual spaces, quotients. 
  • Presentation of algebras using generators and relations. Symmetric and alternating algebras, universal enveloping algebras, the Poincare-Birkhoff-Witt theorem. 
  • Coalgebras and their representations. The fundamental theorem on coalgebras. 
  • Bialgebras, Hopf algebras and their properties. Sweedler notation. Examples, e.g. group algebras. 
  • (Co)actions of Hopf algebras on algebras. Quantum symmetries. 
  • Duality pairing. The Drinfeld double. The Heisenberg algebra. 
  • The Drinfeld-Jimbo and Manin quantum groups. q-calculus and q-deformations. 
  • Quasitriangular structures. The quantum Yang-Baxter equation. Braidings. Self-duality. The discrete Fourier transform. 
  • Cocycles. Drinfeld-Majid twists. Examples from quantum mechanics and representation theory. 

Lecturer

  • Dr Yuri Bazlov

    Dr Yuri Bazlov

    University
    University of Manchester

Bibliography

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Assessment

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Recorded Lectures

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