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.



In what sense do Jones' original subfactors come from quantum SU(2)
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)


