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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?

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This isn't a precise technical answer but the root of unity or not divide is (in ways that can be made precise) analogous to the characteristic zero versus characteristic p divide. Away from the root of unity case, most things are nice; in the root of unity case they go just as badly wrong as Lie algebras in characteristic p. (NB. The interesting shift here is that the characteristic of the actual field underlying $U_{q}(\mathfrak{g})$ isn't in the foreground: the split in behaviour described above happens even for $U_{q}(\mathfrak{g})$ as an algebra over the complex numbers.) –  Jan Grabowski Apr 8 '13 at 15:32

2 Answers 2

Having said that the two have radically different representation theories says already a lot. But first be warned that quantum groups at roots of unity may come in different ways: a beautiful summary was written here Which is the correct version of a quantum group at a root of unity?

Having said so let me add something about the De Concini-Kac form. In such case the quantized enveloping algebra shows a much bigger center. While for $q$ not a root of unity the center is contained in the "diagonal" part $U^0$ (the subalgebra generated by $K$'s elements), for $q^l=1$ the center $Z$ becomes:

  • much bigger; in fact $U_q(\mathfrak g)$ is finite-dimensional over its center;

  • It turns out to be a Hopf subalgebra; being a commutative Hopf algebra it is then the algebra of rational functions on an algebraic group. This turns out to be the algebraic Poisson dual.

  • Since irreps naturally are sent to points of $Spec Z$ one has a map from irreps to the algebraic group $G^\star$ ; such map respect the Poisson structure in the sense that irreps over the same symplectic leaf in $G^\star$ are equivalent. This generalizes the so called orbit method for representations of $\mathfrak g$. Since Lie irreps of $\mathfrak g$ are algebra irreps of $U({\mathfrak g})$ and since this universal enveloping algebra can be seen as a quantization of the natural Poisson structure on $\mathfrak g^\star$, where symplectic leaves are coadjoint orbits;

  • the whole situation is very closely related to irreps of Lie algebras in char $p$ where the same basic features of a big center over whch the whole algebra is finite-dimensional appears (no surprise this latter theory was developed by the same V. Kac, together with Weisefeiler at the beginning of the 70ies).

  • tensor product of irreps behaves again quite differently from the non roots of unity case. In the tensor product of two irreps indecomposables may appear, and in general not much is known about such decompositions. In a sense, from dimension $l$ onwards irreps start to behave more like infinite-dimensional irreps rather than classical finite-dimensional ones.

Lastly, if you wish, for real $q$ you have that the thoery of unitary representations, much as in the classical case, can explain a lot of interesting relations for $q$--special functions; none of which survives (diverging singularities) at $|q|=1$.

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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.

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