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Colloquium October 10

Date: Friday October 10

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Imre Tuba of the Math Dept

Title: Low-dimensional representations of the braid group

Abstract

Ever since Emil Artin first defined the braid group in 1925, it has found applications in such diverse subfields of mathematics and theoretical physics as algebra, low-dimensional topology, operator algebras, quantum mechanics, cryptology, dynamical systems, and quantum computing. Yet, relatively little is known about the representation theory of the braid group. Useful representations have been constructed by Burau, Formanek, Gassner, Jones, and Lawrence among others, and Bigelow recently proved that every braid group has a faithful representation. But the classification of all representations remains open.

I will present joint work with Hans Wenzl on classifying simple representations of the braid group on three strings up to dimension 5. In particular, I will show that the following is almost true: any choice of eigenvalues for the images of the braid generators gives a unique representation. Admittedly a modest start, even this has been a welcome tool in classifying certain tensor categories related to representations of classical quantum groups and in computing categorical dimensions of representations of exceptional quantum groups, all of which appear in cutting-edge quantum mechanics.

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