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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 Virc generated by elements Ln, (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 Ln* = L-n.

• The Temperley-Lieb algebra is the *-algebra TLδ  with generators Ui, (i∈ℤ), relations $$U_i^2 = \delta U_i,\qquad U_iU_{i+1}U_i=U_i,\qquad U_iU_{i-1}U_i=U_i,\qquad U_i U_j=U_j U_i\quad (|i-j|\ge 2)$$ and *-structure Ui* = Ui.

Both Virc and TLδ  depend on a parameter. These are the numbers c and δ ∈ 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. 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:

$$\matrix{ & Case\quad 1. & Case\quad 2. & Case\quad 3. \\ Vir_c\qquad & \quad c\in[1,\infty) & c\in 1-\frac{6}{m(m+1)} for\,\,\,\, m = 2,3,4,5,\ldots\quad & All\quad the \quad rest \\ TL_\delta\qquad & \delta\in[2,\infty) & \delta\in 2\cos\big(\frac\pi n\big)\quad for\quad n = 3,4,5,6,\ldots & All\quad the \quad rest}$$

Namely, both Virc 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 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 Virc to TLδ, or vice-versa?
• Is there any way of linking the values $c\in 1-\frac{6}{m(m+1)}$ and $\delta\in 2\cos(\frac\pi n)$?
• 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.]

• 4 added the positive energy constraint. Without it, the stated facts about Rep(Vir_c) aren't correct.; added 70 characters in body; deleted 4 characters in body

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 Virc generated by elements Ln, (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 Ln* = L-n.

• The Temperley-Lieb algebra is the *-algebra TLδ  with generators Ui, (i∈ℤ), relations $$U_i^2 = \delta U_i,\qquad U_iU_{i+1}U_i=U_i,\qquad U_iU_{i-1}U_i=U_i,\qquad U_i U_j=U_j U_i\quad (|i-j|\ge 2)$$ and *-structure Ui* = Ui.

Both Virc and TLδ  depend on a parameter. These are the numbers c and δ ∈ 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. 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:

$$\matrix{ & Case\quad 1. & Case\quad 2. & Case\quad 3. \\ Vir_c\qquad & \quad c\in[1,\infty) & c\in 1-\frac{6}{m(m+1)} for\,\,\,\, m = 2,3,4,5,\ldots\quad & All\quad the \quad rest \\ TL_\delta\qquad & \delta\in[2,\infty) & \delta\in 2\cos\big(\frac\pi n\big)\quad for\quad n = 3,4,5,6,\ldots & All\quad the \quad rest}$$

Namely, both Virc 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 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 Virc to TLδ, or vice-versa?
• Is there any way of linking the values $c\in 1-\frac{6}{m(m+1)}$ and $\delta\in 2\cos(\frac\pi n)$?
• Are there other algebras exhibiting a similar phenomenon?

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# Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?

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