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General


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 1980, 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.

Semester

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

Hours

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

Timetable

  • Wed 10:05 - 10:55

Prerequisites

  • Essential: undergraduate linear algebra, group theory, ring theory.
  • Advanageous: 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


Yuri Bazlov
Email Yuri.Bazlov@manchester.ac.uk
Phone 01612755816
Photo of Yuri Bazlov


Bibliography


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Assessment



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Recorded Lectures


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