Quantum representations and their algebraic properties
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Thu, 15/11/2012 16:30 |
Søren Fuglede Jørgensen (Aarhus University) |
Junior Geometric Group Theory Seminar  |
St John's |
In this part, I will redefine the
quantum representations for  making no mention of flat
connections at all, instead appealing to a purely combinatorial
construction using the knot theory of the Jones polynomial.
Using these, I will discuss some of the properties of the
representations, their strengths and their shortcomings. One of their
main properties, conjectured by Vladimir Turaev and proved by Jørgen
Ellegaard Andersen, is that the collection of the representations
forms an infinite-dimensional faithful representation. As it is still an
open question whether or not mapping class groups admit faithful
finite-dimensional representations, it becomes natural to consider the
kernels of the individual representations. Furthermore,
I will hopefully discuss Andersen's proof that mapping class groups of
closed surfaces do not have Kazhdan's Property (T), which makes
essential use of quantum representations. |
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