For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions:
• The (universal envelopping algebra of the) Virasoro algebraVirasoro algebra is the *$\star$-algebra Virc$Vir_c$ generated by elements Ln$L_n$, (n∈ℤ$n \in \mathbb{Z}$), 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 Ln* = L-n$L_n^* = L_{-n}$.
• The Temperley-Lieb algebraTemperley-Lieb algebra is the *$\star$-algebra TLδ with$TL_{\delta}$ with generators Ui,$U_i$ (i∈ℤ$i \in \mathbb{Z}$), 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 Virc$Vir_c$ and TLδ depend$TL_{\delta}$ depend on a parameter.
These are the numbers c$c$ and δ ∈ ℝ$\delta \in \mathbb{R}$.
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 Virc and TLδ. 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.
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. The algebra admits faithfulDiscrete series (only) of quotient of Verma modules are unitary representationsand positive energy.
2. It doesn't have faithfulContinuum of Verma modules are unitary representations, but a non-trivial quotient doespositive energy representations.
3. The onlyVerma modules are not 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}$$\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}$
Namely, bothThe parameters Virc$c$ and TLδ exhibit a "discrete series" of non-faithful unitary representations, followed by a continuum of faithful unitary representations. For the discrete series, the parameter should$\delta$ belong to a countable set (discrete series) exhibiting an accumulation point. For higher values of the parameter everything is allowed, and we get faithful representationsfollowed by a continuum.
- Is it pure coincidence that those two algebras exhibit such similar behaviour?
- Is there some natural map from Virc$Vir_c$ to TLδ$TL_{\delta}$, 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?
[Added later: I'm actually not sure that the discrete series of Virc form non-faithful representations...
I'll have to think about that.]
