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"Quantum SU(2) faithfully detects mapping class groups modulo center", Michael Freedman, Kevin Walker, Zhenghan Wang, 2002

Reviewed December 27, 2023

Citation: Freedman, Michael H., Kevin Walker, and Zhenghan Wang. "Quantum SU (2) faithfully detects mapping class groups modulo center." Geometry & Topology 6.2 (2002): 523-539.

Web: https://arxiv.org/abs/math/0209150

Tags: Mathematical, Quantum-groups, Modular-tensor-categories

This paper establishes that the intersection of the kernels of the mapping class group representations associated to U_q(SU(2)) is as small as it could be, as q varies over even-order roots of unity. This is part of a wider programme in quantum topology. Given a modular tensor category, the associated mapping class group representations cannot be faithful. In particular, Dehn twists have infinite order but their images must have finite order. However, given an ADE Lie algebra, the associated mapping class groups can be asymptotically faithful in the sense that the kernel will become arbitrarily small as the deformation parameter "q" becomes large. This is exactly what is shown in this paper for SU(2), modulo elements which are guaranteed to always be in the center.

It is important to note that at the same time this paper was being constructed, a parallel work

> Andersen, Jørgen Ellegaard. "Asymptotic faithfulness of the quantum SU (n) representations of the mapping class groups." Annals of Mathematics (2006): 347-368.

which established similar results for SU(n).