Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity

Question

I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$ works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed everywhere, but I'm having a hard time finding reliable (and clear) sources for representation theory for $q$ being a root of unity.

I've understood that when $q$ is a root of unity, some of the spin $j\in\frac{1}{2}\mathbb{Z}$ representations become reducible while remaining indecomposable. But how do their tensor product representations behave? I've read in some references about the $q$-deformed Clebsch–Gordan coefficients for $q^m\neq1$, but I was unable to find a section dedicated to tensor product representations for $q^m=1$.

Can we still define some sort of Clebsch–Gordan coefficients also when $q^m=1$? I've read something about “removing” representations and defining truncated tensor products, but the statements were confusing... what happens there? I would appreciate some reference suggestions.

Answer 1

The reason you're having trouble finding similarly simple explanations in the root of unity case is that the root of unity case is a lot more complicated. So rather than trying to answer your question directly let me instead give you some key signposts for what to look out for to see why things are complicated.

First, the first definition you look at for a quantum group probably works over the base field $\mathbb{Q}(q)$. This actually means it doesn't make sense to specialize $q$ to any number whatsoever! In order to talk about specializing, you have to pick an "integral form", i.e. you want to work with a ring over say $\mathbb{Q}[q,q^{-1},q+q^{-1}]$ which has the property that it gives you the old quantum group when you base extend. The good news is that so long as $q$ is not a root of unity you can choose any reasonable integral form and you'll end up with the same theory, so you can basically ignore this whole issue. But if you're working at a root of unity you can't ignore this issue. Morever, the integral form that you probably want to use (the Lusztig form) is not the obvious one (in particular it's no longer generated by $E$, $F$, and $K$!).

Second, looking at all representations of the quantum group at a root of unity is too complicated and doesn't look much like what happens at generic $q$. Instead you only want to look at the "nice" representations. The name for these nice representations are "tilting modules." There's an easy elementary definition, namely they're the direct sums of direct summands of tensor products of the defining 2-dimensional representation, however in order to actually prove results about them you need a less elementary definition (in terms of certain filtrations) and you need to see that these two definitions are equivalent.

Third, as you mention, for many applications you don't actually want the category of tilting modules, you actually want to take a quotient of this category (by the "negligible morphisms"). In order to show that this quotient has the properties you want you need to prove some results about the combinatorics of negligible tilting modules. Furthermore, from a concrete viewpoint this quotient is somewhat confusing because the resulting category is no longer the category of representations of any Hopf algebra.

The most elementary way to get at this category is to just ignore quantum groups entirely and directly construct the category you want via the Temperley-Lieb-Jones categories. That is, you just define a category whose objects are strings of dots, whose morphisms are linear combinations of planar arc diagrams modulo planar isotopy and a circle relation, where compositon is stacking and tensor is horizontal disjoint union. Then semismplifying is just taking a quotient setting a certain Jones-Wenzl projection to zero. But then because you don't see the quantum groups anywhere, and so it's much more difficult to generalize beyond $\mathfrak{sl}_2$.