In what sense do Jones' original subfactors come from quantum SU(2) In his paper Index for subfactors [Invent. Math., vol. 72 (1983), pp. 1-26], Vaughan Jones proved his remarkable index rigidity theorem, i.e., the fact that the possible index values for a (type II$_1$) subfactor are precisely those in the set
$$
\{4\cos(\pi/n)^{2}\,:\,n\geq 3\}\cup [4,+\infty].
$$
In particular, he constructed a subfactor with index $\alpha$ for each value $\alpha$ in the "discrete series" $\{4\cos(\pi/n)^{2}\,:\,n\geq 3\}$.
In section 2.4 of the report  https://www.birs.ca/workshops/2014/14w5083/report14w5083.pdf it is stated, with reference to these subfactors, that

The subfactors arising from the discrete series in his [Vaughan Jones']
  article come from SU$_q$(2) at a root of unity.

I would like to know precisely what is meant by this statement.
 A: You can construct a subfactor (under very mild assumptions) from an object $X$ in a rigid C*-tensor category, by taking the limit of inclusions 
$$
{\rm End}(X^{\otimes n}) \simeq \{ \iota_X \otimes T \mid T \in {\rm End}(X^{\otimes n})\} \subset {\rm End}(X^{\otimes n+1})
$$
as $n \to \infty$. One good entry point is the Longo-Roberts paper [LR97], I guess. In particular, you get Jones's subfactors from the "half spin" object in ${\rm Rep}({\rm SU}_q(2))$ for root of unity $q$.
[LR97] R. Longo and J. E. Roberts, A theory of dimension, K-Theory 11 (1997), no. 2, 103–159. MR 1444286 (98i:46065)
A: The fusion category $\mathcal{C}_\ell$ of unitary highest weight projective representations of level $\ell$  of the loop group $LSU(2)$ is equivalent to ${\rm Rep}({\rm SU}_q(2))$ with $q = e^{\frac{i \pi}{\ell + 2}}$ (see this paper, first paragraph p5).   
Now, for any simple object $\rho$ of $\mathcal{C}_\ell$ (characterized by its spin $i \le \ell/2$), there is a Jones-Wassermann subfactor (see this Jones' survey Section 6, or also this answer): $$ \rho (L_I)'' \subseteq \rho (L_{I^c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}$ with $m=\ell + 2$ and $p=2i+1$.  
At spin $1/2$, it is exactly the Jones's original subfactor of index $4cos^2(\frac{\pi}{m})$.
