For those who aren't familiar with the [Virasoro][1] or [Temperley-Lieb][2] algebras, I include some definitions:

&bull; The (universal envelopping algebra of the) <b>Virasoro algebra</b> is the \*-algebra <i>Vir<sub>c</sub></i> generated by elements <i>L<sub>n</sub></i>, (n&isin;&#8484;), 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 \*-structure <i>L<sub>n</sub></i>\* = <i>L<sub>-n</sub></i>.

&bull; The <b>Temperley-Lieb algebra</b> is the \*-algebra <i>TL<sub>&delta;</sub></i>&nbsp; with generators <i>U<sub>i</sub></i>, (i&isin;&#8484;)</i>, 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 <i>Vir<sub>c</sub></i> and <i>TL<sub>&delta;</sub></i>&nbsp; depend on a parameter.
These are the numbers <i>c</i> and &delta; &isin; &#8477;.<br>
<hr>
Let's call a representation &rho; of a \*-algebra on a Hilbert space <i>unitary</i> if &rho;(x\*) = &rho;(x)\*. We are interested in the unitary representations of 
<i>Vir<sub>c</sub></i> and <i>TL<sub>&delta;</sub></i>. In the case of the Virasoro algebra, we further restrict to positive energy representations, i.e., reps in which the spectrum of <i>L</i><sub>0</sub> is positive.
Depending on the value of the parameters <I>c</i> and &delta;, three things can happen:  

**1.** The algebra admits faithful unitary representations.   
**2.** It doesn't have faithful unitary representations, but a non-trivial quotient does.  
**3.** The only unitary representation is the zero representation.  

Now here's the striking thing:

$\begin{array}{c|c|c|c|c|c|c}
    & \text{Case 1.} & \text{Case 2.} & \text{Case 3.}  \newline 
                             \hline
Vir_c & c \in [1,\infty) &c \in \{ 1-\frac{6}{m(m+1)} \text{ for } m = 2,3,4,5,\ldots \}  & \text{non-unitary}  \newline
                             \hline
TL_\delta & \delta \in [2,\infty)  &\delta\in \{ 2\cos\big(\frac\pi m\big)\quad for\quad m = 2,3,4,5,6,\ldots \} & \text{non-unitary}   
\end{array}$    


Namely, both <i>Vir<sub>c</sub></i> and <i>TL<sub>&delta;</sub></i> exhibit a "discrete series" of non-faithful unitary representations, followed by a continuum of faithful unitary representations. For the discrete series, the parameter should belong to a countable set exhibiting an accumulation point. For higher values of the parameter everything is allowed, and we get faithful representations.

> - Is it pure coincidence that those two algebras exhibit such similar behaviour?    
> - Is there some natural map from <i>Vir<sub>c</sub></i> to <i>TL<sub>&delta;</sub></i>, 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?

 

<hr>
[<i>Added later:</i> I'm actually not sure that the discrete series of <i>Vir<sub>c</sub></i> form non-faithful representations...<br> I'll have to think about that.]

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