The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$ I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for example this question.
But we get $U(\mathfrak{sl}_2) \otimes \mathbb{Z}/2$. This fact goes on to create many problems, such as the effective doubling of the number of representations, half of which are generally regarded as "uninteresting".
It seems to me that finding a presentation of $U_q(\mathfrak{sl}_2)$ would be of great benefit. However I can't seem to find any discussion of this anywhere. Can anybody explain what is going on here? Is the double cover somehow a 'necessary consequence' of quantization? Is there some reason why people felt it best to stick with the double cover approach?
 A: Well, i am not sure if this is what the OP is looking for but here is an heuristic method for computing the limit, avoiding the use of another algebra defined at $q=1$ and thus bypassing the "double cover problem" (mentioned in the OP):
Let us use the standard presentations in terms of generators and relations,

*

*$U\big(sl(2)\big)$ is:
$\ \ \ [H,X]=2X$, $[H,Y]=-2Y$, $[X,Y]=H$
whereas

*$U_q\big(sl(2)\big)$ is:
$\ \ KK^{-1}=K^{-1}K=1$, $KE=q^2EK$, $KF=q^{-2}FK$, $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$
The problem is that we cannot just plug-in $q=1$ in the above relations because the limit is indeterminate. In order to work around this problem (essentially following Drinfeld's original approach), let us use the "change of variables":
$$
q=e^{h/2} \ \ \ \ K=q^H=e^{hH/2}
$$
These imply: $$\frac{dK}{dh}=\frac{HK}{2}, \ \ \ \ \ \ \ \ \ \ \lim_{h\to 0}K=1$$
Now
$$
[K,E]=KE-EK=(q^2-1)EK=(e^h-1)EK
$$
Differentiating the last relation wrt to $h$ gives:
$$
\frac{1}{2}[HK,E]=e^hEK+(e^h-1)E\frac{h}{2}K
$$
and finally take the limit of the above while $h\to 0$ to get:
$$[H,E]=2E$$
With the correspondence $E\leftrightarrow X$ and $F\leftrightarrow Y$ (the $[K,F]$ relation is treated in a similar manner) these give the first two relations of $U\big(sl(2)\big)$.
Now the third relation is written as
$$
[E,F]=\frac{K-K^{-1}}{q-q^{-1}}=\frac{e^{hH/2}-e^{-hH/2}}{e^{h/2}-e^{-h/2}}
$$
Take the limit of both sides at $h\to 0$, using Del' Hospital in the rhs:
$$
[E,F]=\lim_{h\to 0}\frac{\frac{HK}{2}-(-\frac{HK}{2})}{\frac{1}{2}e^{h/2}-(-\frac{1}{2}e^{-h/2})}=H
$$
which gives the third relation describing the multiplication of the $U\big(sl(2)\big)$ algebra as the $q\to 1$ limit of the multiplication of the $U_q\big(sl(2)\big)$ algebra.
Finally, the coalgebra structure (comultiplication, counity) and the antipode  limits can be handled in a similar manner to conclude that the $q\to 1$ limit of the Quantum group $U_q\big(sl(2)\big)$ is the Hopf algebra $U\big(sl(2)\big)$.
Edit: Regarding your last question:

Is there some reason why people felt it best to stick with the double cover approach?

i would say that (contrary to the remark in the OP:

This fact goes on to create many problems, such as the effective doubling of the number of representations ...)

this "doubling" of the representations is the reason behind the double-cover approach in the sense that this provides a more direct analogy between the representations of $U_q\big(sl(2)\big)$ and the representations of $U\big(sl(2)\big)$: The representation theory of $U_q$ includes two classes of highest weight modules, parameterized by the positive integers and the sign $\epsilon =\pm 1$, unlike the rep theory of $U$ which includes a single class of highest weight modules parameterized by the positive integers.
(This happens genererally for $q$ being either a root of unity or not, although for the root of unity case things are a little more subtle i.e. there are other classes of reps as well, an upper bound in their dim etc; but this is irrelevant to the rest).
So, when taking the $q\to 1$ limit of $U_q$, its $\epsilon =1$ reps give the usual $U$-highest weight modules while its $\epsilon =-1$ reps correspond to the "double" class of reps of $U(\mathfrak{sl}_2) \otimes \mathbb{Z}/2$ (since the last algebra can be viewed as an extension of $U$ by adjoining a single, central generator satisfying $g^2=1$).
