For those who aren't familiar with the [Virasoro][1] or [Temperley-Lieb][2] 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][3] 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?


  [1]: http://en.wikipedia.org/wiki/Virasoro_algebra
  [2]: http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra
  [3]: http://en.wikipedia.org/wiki/Verma_module