TITLE: Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?
QUESTION [14 upvotes]: Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If I have understood, two reasons are:
(i) The representation theory of the two types is very different.
(ii) At roots of unity, one can quotient $U_q(\frak{g})$ to get a quasi-triangular Hopf algebra.
Do there exist other important ways in which these two types differ?

REPLY [4 votes]: Your first statement is certainly right on the money.  Away from roots of unity the representation theory behaves essentially the same as the representation theory of the classical Lie algebra, while at roots of unity it becomes nonsemisimple and behaves quite like the more complicated rep theory of Lie algebras over finite fields.  Your second statement is a little off though. $U_q({\mathfrak{g}})$ is quasitriangular (up to issues of completion) and indeed ribbon for all values of $q,$ and thus gives things like link invariants.  In fact what happens at roots of unity is that a quotient of a subcategory of the representation theory forms a modular category, which gives a TQFT and a three manifold invariant, which seems to replicate the theory one expects heuristically to get from Chern-Simons quantum field theory.  So the roots of unity are particularly challenging to understand and particularly interesting in terms of representation theory, topology and physics.