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 *-algebra Vir_c generated by elements L_n, (n∈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 *-structure L_n* = L_-n.

• The Temperley-Lieb algebra is the *-algebra TL_δ  with generators U_i, (i∈ℤ), 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_δ  depend on a parameter. These are the numbers c and δ ∈ ℝ.

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Let's call a representation ρ of a \*-algebra on a Hilbert space unitary if ρ(x\*) = ρ(x)\*. We are interested in the unitary representations of Vir_c and TL_δ. In the case of the Virasoro algebra, we further restrict to positive energy representations, i.e., reps in which the spectrum of L₀ is positive. Depending on the value of the parameters c and δ, 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 Vir_c and TL_δ 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 Vir_c to TL_δ, 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?

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[Added later: I'm actually not sure that the discrete series of Vir_c form non-faithful representations...
I'll have to think about that.]