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"Quantum Groups at Roots of Unity and Modularity", Stephen Sawin, 2006

Reviewed August 31, 2023

Citation: Sawin, Stephen F. "Quantum groups at roots of unity and modularity." Journal of Knot Theory and Its Ramifications 15.10 (2006): 1245-1277.

Web: https://arxiv.org/abs/math/0308281

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

This paper properly establishes the conditions for which the representation category of a quantum group will be modular.

The author puts it beautifully: "To each simple Lie algebra there was associated a quantum group. By understanding the representation theory of this quantum group at roots of unity in an analogous fashion to Reshetikhin and Turaev's work on sl2, one could presumably show that this representation theory formed a modular tensor category, and thus construct an invariant of links and three-manifolds, presumably the one Witten associated to the corresponding compact, simple Lie group".

This seems like an obvious paper to write. The question might be what took so long? Well, again in the author's words: "Much of the obstacle to the complete resolution of this problem appears to be faulty communication between those working on the algebraic questions and the topologists and mathematical physicists interested in the link and three-manifold invariants... Thus remarkably, there is no proof in the literature that from any quantum group at roots of unity one can construct three-manifold invariants, or even semisimple ribbon categories, though of course these facts are widely understood to be true."

This is exactly my sort of paper - it pulls together lots of literature, gives a nice summary and discussion of what's known, and then fills in an obvious gap.