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?