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André Henriques
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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?

For those who aren't familiar with the [Virasoro][1] or [Temperley-Lieb][2] 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. 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? [1]: http://en.wikipedia.org/wiki/Virasoro_algebra [2]: http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra
  • André Henriques
    • 43.2k
    • 5
    • 130
    • 264