For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity case. The later being semi-simple, the former not. I wonder how the $q=-1$ case look like. Is it in any sense "less badly behaved" than the complex root of unity case? As a concrete question, for $q=-1$, how far is the category of modules from being semi-simple.

  • $\begingroup$ As someone who constantly struggles with sign issues, and otherwise doesn't know from representations of quantum groups, I would expect the $q = -1$ case to be more complicated, not less. $\endgroup$
    – LSpice
    Oct 30, 2018 at 14:23
  • $\begingroup$ Also, although I guess anyone in a position to answer will know what you mean, I find the construction confusing ("$q$ a root of unity, and $q$ not a root of unity …. the lat[t]er being semi-simple, the former not" sure looked like 'semi-simple' goes with "$q$ a root of unity", until I read carefully and noticed 'later'). $\endgroup$
    – LSpice
    Oct 30, 2018 at 14:40
  • 2
    $\begingroup$ As Calvin says you have to be careful about conventions, or else $q=\pm 1$ won't make sense. That said, very vaguely $q=-1$ is a lot like $q=1$ but perhaps with a slightly "super" flavor (i.e. some weird signs in the braiding). $\endgroup$ Feb 24, 2019 at 20:24

1 Answer 1


It depends exactly on your conventions, but in a form of the quantum group with a relation along the lines of $$ [E, F] = \frac{K - K^{-1}}{q - q^{-1}} $$ won't be well-defined at $ q = \pm 1$. To get phenomena associated with second roots of unity, one sets $q$ to be a fourth root of unity.

As an example of what I'm referring to, if $q$ is a primitive $l$th root of unity for $l$ odd, you get cyclic $l$-dimensional representations, while if $q$ is a primitive $2r$th root of unity, you get cyclic $r$-dimensional representations. For this reason, a lot of literature excludes the $q = \pm 1$ case.

I'm not well-versed in the more combinatorial aspects of quantum groups, and I know that there are other presentations with different behaviors over different coefficient rings. You may want to find someone who knows more to expand on this answer.


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