For those who aren't familiar with the [Virasoro](http://en.wikipedia.org/wiki/Virasoro_algebra) or [Temperley-Lieb](http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra) 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](http://en.wikipedia.org/wiki/Verma_module) 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?
> 
> 
> 



Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?
**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](http://archive.numdam.org/ARCHIVE/SB/SB_1994-1995__37_/SB_1994-1995__37__251_0/SB_1994-1995__37__251_0.pdf).  

- 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](http://arxiv.org/abs/1010.0078) (2010)  

- S. Palcoux, *Neveu-Schwarz and operators algebras II : Unitary series and characters*, [arXiv:1010.0077](http://arxiv.org/abs/1010.0077) (2010)  

- S. Palcoux, *Neveu-Schwarz and operators algebras III : Subfactors and Connes fusion*, [arXiv:1010.0076](http://arxiv.org/abs/1010.0076) (2010)  

- V. Toledano Laredo, *Fusion of Positive Energy Representations of LSpin(2n)*, thesis, Cambridge 1997, [arXiv:math/0409044](http://arxiv.org/abs/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](http://link.springer.com/article/10.1007/s002220050253).  

- A. J. Wassermann, *Kac-Moody and Virasoro algebras*, 1998, [arXiv:1004.1287](http://arxiv.org/abs/1004.1287) (2010)  

- A. J. Wassermann, *Subfactors and Connes fusion for twisted loop groups*, [arXiv:1003.2292](http://arxiv.org/abs/1003.2292) (2010) 


