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As for the loop algebra, there are $\mathfrak{Vir}_{I}$ and $\mathcal{N}_{pq}^{m}(I)$ generated by $\pi_{pq}^{m}(\mathfrak{Vir}_{I})$.
We obtain the Jones-Wassermann subfactor :
$$\mathcal{N}_{pq}^{m}(I) \subset \mathcal{N}_{pq}^{m}(I^{c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}.\frac{sin^{2}(q\pi/m)}{sin^{2}(\pi/m)}$. Its principal graph is given by the fusion rules :
$$H_{pq}^{m} \boxtimes H_{p'q'}^{m} = \bigoplus_{(i'',j'') \in \langle i,i' \rangle_{\ell} \times \langle j,j' \rangle_{\ell + 1} }H_{p''q''}^{m}$$ with $p=2i+1, q=2j+1, p'=2i'+1, ..., m=\ell+2$

As for the loop algebra, there are $\mathfrak{Vir}_{I}$ and $\mathcal{N}_{pq}^{m}(I)$ generated by $\pi_{pq}^{m}(\mathfrak{Vir}_{I})$.
We obtain the Jones-Wassermann subfactor :
$$\mathcal{N}_{pq}^{m}(I) \subset \mathcal{N}_{pq}^{m}(I^{c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}.\frac{sin^{2}(q\pi/(m+1))}{sin^{2}(\pi/(m+1))}$. Its principal graph is given by the fusion rules :
$$H_{pq}^{m} \boxtimes H_{p'q'}^{m} = \bigoplus_{(i'',j'') \in \langle i,i' \rangle_{\ell} \times \langle j,j' \rangle_{\ell + 1} }H_{p''q''}^{m}$$ with $p=2i+1, q=2j+1, p'=2i'+1, ..., m=\ell+2$

• $\mathcal{L} \Omega = \ell \Omega$ with $\ell \in \mathbb{N}$ and $\Omega$ the vacuum vector.

• $i \in \frac{1}{2}\mathbb{N}$ and $i \le \frac{\ell}{2}$, the spin (related to the irreducible representation $V_{i}$ of $\mathfrak{g}$)

• $\mathcal{L} \Omega = \ell \Omega$ with $\ell \in \mathbb{N}$ the level, and $\Omega$ the vacuum vector.

• $i \in \frac{1}{2}\mathbb{N}$ and $i \le \frac{\ell}{2}$, the spin (related to the irreducible representation $V_{i}$ of $\mathfrak{g}$)

Temperley-Lieb case :
If $i=1/2$ then index=$\frac{sin^{2}(2\pi/(\ell+2))}{sin^{2}(\pi/(\ell+2))} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{\ell+2})$ and the principal graph is $A_{\ell+1}$.
In this case, the subfactors are known to be completely classified by their principal graph.
The subfactor planar algebra it generates is the Temperley-Lieb planar algebra $TL_{\delta}$.

Temperley-Lieb case :
If $(p,q) = (2,1)$, index$=\frac{sin^{2}(2\pi/m)}{sin^{2}(\pi/m)} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{m})$ and the principal graph is $A_{m-1}$.
As above, the subfactor planar algebra is Temperley-Lieb $TL_{\delta}$.

Temperley-Lieb case (with $\ell \ge 1$) :
If $i=1/2$ then index=$\frac{sin^{2}(2\pi/(\ell+2))}{sin^{2}(\pi/(\ell+2))} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{\ell+2})$ and the principal graph is $A_{\ell+1}$.
In this case, the subfactors are known to be completely classified by their principal graph.
The subfactor planar algebra it generates is the Temperley-Lieb planar algebra $TL_{\delta}$.

Temperley-Lieb case (with $m \ge 3$) :
If $(p,q) = (2,1)$, index$=\frac{sin^{2}(2\pi/m)}{sin^{2}(\pi/m)} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{m})$ and the principal graph is $A_{m-1}$.
As above, the subfactor planar algebra is Temperley-Lieb $TL_{\delta}$.