Complex semisimple quantum groups and representation theory

Christian Voigt (Glasgow)

Abstract:

In these lectures I will give an introduction to the theory of complex semisimple quantum groups. These quantum groups are deformations of classical complex semisimple Lie groups, and an instructive special case is the quantum Lorentz group, which is obtained by deforming the group SL(2,C) in a suitable way. After giving some general background on quantum groups and their representations, the first aim will be to describe the construction of the C*-algebras associated with complex quantum groups. I will then focus on some aspects of their representation theory, indicating similarities/differences to the situation for classical complex groups, and illustrating the results in the case of the quantum Lorentz group. Time permitting, I will explain how this relates to K-theory and the Baum-Connes conjecture.

References:

No background on quantum groups will be assumed, but some familiarity with Woronowicz's theory of compact quantum groups should be helpful:

• Maes, Van Daele - Notes on compact quantum groups. [MR1645264]
• Neshveyev, Tuset - Compact quantum groups and their representation categories. [MR3204665]
• Woronowicz - Compact quantum groups. [MR1616348]

For an introduction to the quantum Lorentz group SLq(2, C) and its representation theory see:

• Podles, Woronowicz - Quantum deformation of Lorentz group. [MR1059324]
• Pusz, Woronowicz - Unitary representations of quantum Lorentz group. [MR1306544]

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