Quantum Groups

Join Ryan Aziz from Queen Mary College, University of London for this week's seminar.

  • Thu 22 Feb 18

    14:00 - 15:00

  • Colchester Campus

    Room EBS 2.1

  • Event speaker

    Ryan Aziz, Postgraduate Research Student, Queen Mary College, University of London

  • Event type

    Lectures, talks and seminars
    Department of Mathematical Sciences Seminar Series

  • Event organiser

    Mathematical Sciences, Department of

  • Contact details

    Dr Andrew Harrison

Quantum groups were first discovered independently by Drinfeld and Jimbo in the 1980s in their study of quantum integrable system. Although we do not have a single definition of quantum groups, one would agree that it is a deformation of some algebras by some rules involving quantum constant q, where at the limit q -> 1, we recover the classical cases. Although we will not discuss it in detail, quantum groups are used in wide range of topics. For example, it is one of main ingredient to study non-commutative geometry (or quantum geometry) and according to A. Connes, it might give us a potential proof of Riemann Hypothesis. Quantum groups are also studied as a model to quantum spacetime and quantum gravity, i.e. the ultimate problem in physics: The Theory of Everything.

In this talk, Mr Aziz would like to introduce this fascinating topic. Mr Aziz will talk shortly about quantum groups motivated from quantum physics. Quantum groups have self-duality which is why some people think that quantum groups are some class of (ordinary) Hopf algebras. But the focus of this talk is to introduce Braided Hopf algebras (first defined by Majid, 1990s), which is a Hopf algebra living in braided monoidal category. Mr Aziz will then discuss some of its basic properties.

To summarise, Mr Aziz will talk shortly about double bosonisation of Majid, and the dual version. The terms come from physics, but it is a construction of (dual) quantum groups associate with braided Hopf algebras. Using this method we are able to construct all families of standard quantum groups of complex semisimple Lie algebras following Dynkin diagram, and also non-standard quantum groups.

[1] R. Aziz and S. Majid, Co-double Bosonisation and Dual Basis of cq[SL2] and cq[SL3], arXiv :
1703.03456v2 (2017) 44pp.
[2] S. Majid, Foundation of Quantum Groups, Cambridge University Press (2000) 640pp.