I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I know that when $q$ is a primitive root of unity (say, not equal to 1 or -1) then there are two irreducible representations in each dimension strictly smaller than the degree of $q$ and an infinite family of them in the dimension equal to this degree. Now, I've often heard people say that there is however some sort of finiteness to its category of representations because all the rep's of dimension $deg(q)$ are "of quantum dimension zero". The only notion of "quantum dimension" I know of is in pivotal tensor categories (which the aforementioned are) by taking the trace of identity morphisms (or the pivotal structure maps if you will) but I don't see why this dimension should vanish for these objects. Am I interpreting this right but just am too blind to see it?

Edit: Suppose $d=min(n\in \mathbb{N}:q^n=1)$ then by $deg(q)$ I mean the number $d$ if $d$ is odd and $d/2$ if $d$ is even. An example of the context I'm talking about would be the paper Topological invariants from non-restricted quantum groups as I'm trying to deal with these Turaev-Viro-type models which should be formulated over fusion categories (so in particular categories with finitely many simple objects) so that the sum over dimensions in their definition does not diverge. However, this is unfortunately all hear-say my advisors have thrown at me making it rather difficult for me to actually find reliable sources.

  • $\begingroup$ Can you add a little more information about the sources you are quoting? Most important, can you explain what you mean by "strictly smaller than the degree of $q$"? I'm not sure what "degree of $q$" is here. In one interpretation, the representation theory should mimic the modular theory, but perhaps your quantum dimension zero corresponds to non-restricted representations. $\endgroup$ – Jim Humphreys Dec 6 '15 at 20:01
  • $\begingroup$ Oh absolutely, I'm sorry if I've been unclear. I made a quick edit, I hope this clarifies it. $\endgroup$ – Nephry Dec 6 '15 at 20:23
  • $\begingroup$ I've only approached these things from the viewpoint of representation theory in prime characteristic, but you might get some insight from the papers of Henning Andersen on quantum groups, tilting modules, fusion rules. One of these which is accessible online is here: projecteuclid.org/euclid.cmp/1104272854 (see for example Theorem 3.21). Andersen is looking mainly at restricted representations in charactreristic $p$, but for non-restricted representations in rank 1 there are many Steinberg-type representations having dimension $p$. $\endgroup$ – Jim Humphreys Dec 6 '15 at 21:59

Yes, you are interpreting it right. For $U_q(\mathfrak{sl}_2)$, there's a very concrete interpretation of the quantum dimension: it's the trace (in the usual sense) of the element $K$. So, on the Weyl module of highest weight $n$ (the representation generated by $v$ with the relation $Kv=q^nv$ and $F^{(n+k)}v=E^{(k)}v=0$ for all $k>0$), the quantum dimension is $\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}=q^n+q^{n-2}+\cdots+q^{-n}$. So, if $q^{2n+2}=1$, then this representation has quantum dimension 0.

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  • $\begingroup$ Is there an easy way to see why the quantum dimension is the trace of $K$? Why $K$? For $U_q (sl_3)$ it seems to be the trace of $K_{\alpha_1}K_{\alpha_2}$ but I'm not sure how to do this for $U_q(sl_n)$ in general $\endgroup$ – Logan Tatham Apr 22 '18 at 21:40
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    $\begingroup$ @LoganTatham In general, it's image of $K$ under the principal embedding of $\mathfrak{sl}_2$ (often denoted $K_{\rho}$); you're noting that in $\mathfrak{sl}_3$, we have $\rho=\frac 12(\alpha_1+\alpha_2)$. $\endgroup$ – Ben Webster Apr 23 '18 at 19:27

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