# 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.

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)

• Thanks for your answer! I have a follow-up question: When I see the symbol SU$_q$(2), I think of the compact quantum group (with parameter $q$ between $-1$ and $1$). What exactly is meant by this symbol when $q$ is a root of unity (in the context of your answer)? – hetairoi22 Mar 5 '17 at 10:31
• I think it's better to think of ${\rm Rep}({\rm SU}_q(2))$ as a single thing rather than trying to figure out what ${\rm SU}_q(2)$ is separately—much like there is only the C*-algebra $C({\rm SU}_q(2))$ for positive $q$. There are several easy characterizations of this category: Kazhdan–Wenzl paper (though it seems that the case of SL(2) was due to T. Kerler referenced there), and Chapter XII of Turaev's "Quantum invariants" book. But if you really want to see intermediate Hopf algebra, there's Wenzl's paper "$C^*$ tensor categories from quantum groups" and you can follow its references. – Makoto Yamashita Mar 5 '17 at 19:14

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 $4cos^2(\frac{\pi}{m})$.

• Thanks for your answer! I was wondering if you could clarify something: In what sense are the two categories you mention equivalent? – hetairoi22 Mar 5 '17 at 10:34
• They are equivalent as fusion categories. You can see the reference that I have added to the answer. It is stated there but not explained in details (the references in this reference could help). Note that the notion of "affine" in this reference means that we consider "projective" representations. – Sebastien Palcoux Mar 5 '17 at 18:12