TITLE: Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?
QUESTION [41 upvotes]: For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions:
• The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra $Vir_c$ generated by elements $L_n$, ($n \in \mathbb{Z}$), subject to the relations
$$
[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0},
$$
and with $\star$-structure $L_n^* = L_{-n}$.
• The Temperley-Lieb algebra is the $\star$-algebra $TL_{\delta}$ with generators $U_i$ ($i \in \mathbb{Z}$) and relations :  

$U_i^2 = \delta U_i$   and $*$-structure $U_i^* =  U_i$.
$U_iU_{i+1}U_i=U_i$  and $U_iU_{i-1}U_i=U_i$   
$U_i U_j=U_j U_i$  for  $|i-j|\ge 2$    

Both $Vir_c$ and $TL_{\delta}$ depend on a parameter.
These are the numbers $c$ and $\delta \in \mathbb{R}$.  

Let's call a representation $\rho$ of a $\star$-algebra on a Hilbert space unitary if $\rho(x^*)=\rho(x)^*$.
We are interested in the unitary representations of $Vir_c$ and $TL_{\delta}$. In the case of the Virasoro algebra, we further restrict to positive energy representations, i.e., unitary representations in which the spectrum of $L_0$ is positive.
Depending on the value of the parameters $c$ or $\delta$, three things can happen:  
1. Discrete series (only) of quotient of Verma modules are unitary and positive energy.
2. Continuum of Verma modules are unitary, positive energy representations.
3. The Verma modules are not unitary.    
Now here's the striking thing:
$\begin{array}{c|c|c|c|c|c|c}
    & \text{Discrete series} & \text{Continuum} & \text{Others}  \newline 
                             \hline
Vir_c & c \in \{ 1-\frac{6}{m(m+1)}   \vert   m = 2,3,4 \ldots \}  &c \in [1,\infty) & \text{non-unitary}  \newline
                             \hline
TL_\delta & \delta\in \{ 2\cos\big(\frac\pi m\big)\quad  \vert  \quad m = 2,3,4 \ldots \}  &\delta \in [2,\infty) & \text{non-unitary}   
\end{array}$    
The parameters $c$ and $\delta$ belong to a countable set (discrete series) exhibiting an accumulation point, followed by a continuum. 


Is it pure coincidence that those two algebras exhibit such similar behaviour?    
Is there some natural map from $Vir_c$ to $TL_{\delta}$, or vice-versa?  
Is there any way of linking the values $c\in 1-\frac{6}{m(m+1)}$ and $\delta\in 2\cos(\frac\pi m)$?  
Are there other algebras exhibiting a similar phenomenon?

REPLY [16 votes]: Overview of an explanation :  
Jones-Wassermann subfactors for the loop algebra :
Let $\mathfrak{g} = \mathfrak{sl}_{2}$ be the Lie algebra, $L\mathfrak{g}$ its loop algebra and $\mathcal{L}\mathfrak{g} = L\mathfrak{g} \oplus \mathbb{C}\mathcal{L}$ the central extension :
 $$[X^{a}_{n},X^{b}_{m}] = [X^{a},X^{b}]_{m+n} + m\delta_{ab}\delta_{m+n}\mathcal{L}$$  with $(X^{a})$ the basis of $\mathfrak{g}$. 
  The unitary highest weight representations of $\mathcal{L}\mathfrak{g}$ are $(H_{i}^{\ell},\pi_{i}^{\ell})$ with : 

$\mathcal{L} \Omega = \ell \Omega$ 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 $\mathfrak{g}$)   

Let $I \subset \mathbb{S}^{1}$ an interval, and $\mathcal{L}_{I}\mathfrak{g}$ the local Lie algebra generated by $(X^{a}_{f})$ with :   

$f(\theta) = \sum \alpha_{n}e^{in\theta}$ and $f \in C^{\infty}_{I}(\mathbb{S}^{1})$
$X^{a}_{f} = \sum \alpha_{n}X^{a}_{n}$    

Let $\mathcal{M}_{i}^{\ell}(I)$ be the von Neumann algebra generated by $\pi_{i}^{\ell}(\mathcal{L}_{I}\mathfrak{g})$.
We obtain the Jones-Wassermann subfactor :
$$\mathcal{M}_{i}^{\ell}(I) \subset \mathcal{M}_{i}^{\ell}(I^{c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}$ with $m=\ell + 2$ and $p=2i+1$.
Its principal graph is given by the fusion rules :
$$H_{i}^{\ell} \boxtimes H_{j}^{\ell} = \bigoplus_{k \in \langle i,j \rangle_{\ell}}H_{k}^{\ell}$$ with $\langle a,b \rangle_{n} = \{c=\vert a-b \vert, \vert a-b \vert+1,... \vert c \le a+b , a+b+c \le n \}$
Let $\mathcal{R}_{\ell}$ be the fusion ring generated.  
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 $TL_{\delta}$. 
Jones-Wassermann subfactors for the Virasoro algebra :
Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension:
 $$
[L_m,L_n]=(m-n)L_{m+n}+\frac{C}{12}(m^3-m)\delta_{m+n,0},
$$ 
Its discrete series representations are $(H_{pq}^{m})$ with : 

$C\Omega = c_{m} \Omega$ with $c_{m}= 1-\frac{6}{m(m+1)}$ for $m=2,3,...$
$L_{0} \Omega = h^{pq}_{m} \Omega$ with $h^{pq}_{m} = \frac{[(m+1)p-mq]^{2}-1}{4m(m+1)}$ with $1 \le p \le m-1$ and $1 \le q \le p $  

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 $p=2i+1, q=2j+1, p'=2i'+1, ..., m=\ell+2$   
Let $\mathcal{T}_{m}$ be the fusion ring they generate, it's an easy quotient of  $\mathcal{R}_{\ell} \otimes_{\mathbb{Z}} \mathcal{R}_{\ell+1}$, with $\mathcal{R}_{\ell}$ the fusion ring obtained above for the loop algebra.
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 $TL_{\delta}$. 

$\rightarrow$ We obtain the natural maps $c \leftrightarrow \delta$
  and  $\mathfrak{Vir}_{c} \leftrightarrow TL_{\delta}$ that you
  expected.

Generalizations for similar phenomenon :
Here is a list of possibilities : 

take $i$ other than $1/2$ or $(p,q)$ other than $(2,1)$    
take $\mathfrak{g}$ other than $\mathfrak{sl}_{2}$  
take the continuous series  
take a $N$-super-symmetric extension of $\mathfrak{Vir}$ : $N=1$ for the Neveu-Schwarz and Ramond algebras.     

References :
-  V.F.R. Jones, Fusion en algèbres de von Neumann et groupes de lacets (d'après A. Wassermann),  Séminaire Bourbaki, Vol. 1994/95.  Astérisque  No. 237  (1996), Exp. No. 800, 5, 251--273.
- T. Loke, Operator algebras and conformal field theory for the discrete series representations of $\textrm{Diff}(\mathbb{S}^{1})$, thesis, Cambridge 1994.
- S. Palcoux, Neveu-Schwarz and operators algebras I : Vertex operators superalgebras, arXiv:1010.0078 (2010)
- S. Palcoux, Neveu-Schwarz and operators algebras II : Unitary series and characters, arXiv:1010.0077 (2010)
- S. Palcoux, Neveu-Schwarz and operators algebras III : Subfactors and Connes fusion, arXiv:1010.0076 (2010)
- V. Toledano Laredo, Fusion of Positive Energy Representations of LSpin(2n), thesis, Cambridge 1997, arXiv:math/0409044 (2004)
- R. W. Verrill, Positive energy representations of $L^{\sigma}SU(2r)$ and orbifold fusion.  thesis, Cambridge 2001.
- A. J. Wassermann, Operator algebras and conformal field theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994),  966--979, Birkhuser, Basel, 1995.
- A. J. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of ${\rm LSU}(N)$ using bounded operators. Invent. Math.  133  (1998),  no. 3, 467--538.
- A. J. Wassermann, Kac-Moody and Virasoro algebras, 1998, arXiv:1004.1287 (2010)
- A. J. Wassermann, Subfactors and Connes fusion for twisted loop groups, arXiv:1003.2292 (2010)