## 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 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.]

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Is it clear that this isn't the generic expected behavior for objects of this type? – Qiaochu Yuan Jul 17 2011 at 23:55
@Qiaochu: What do you mean by "objects of this type"? – André Henriques Jul 18 2011 at 0:27
Maybe Qiaochu's question is the following. As physicists, we're very used to the following phenomenon: let $V(x)$ be a potential energy function which bounded below and has finitely many local minima (I probably can relax something). Then the spectrum of the Hamiltonian $(i\hbar\frac{\partial}{\partial x}) + V(x)$ has a discrete part, roughly corresponding to valleys in the graph of $V$, and a continuous part, starting near the tops of the mountains and going higher. Here the role of $c,\delta$ is played by the energy $E$ = eigenvalue. So your remarked upon behavior is not a priori surprising. – Theo Johnson-Freyd Jul 18 2011 at 11:40
@Theo: In your example, is V(x) is a bounded function? (if not: what do you mean by "top of the mountains"?). What is the simplest example of a function V(x) that exhibits the kind of behavior that you describe? – André Henriques Jul 18 2011 at 12:36
@André, something like this: etsf.eu/system/files/born-oppenheimer-m.png -- as one gets closer to the "top of the well" the energy levels are getting finer and eventually they become continuous (it depends on the precise profile whether there is a finite or infinite number of the discrete levels though). I am not sure what is the picture of but qualitatively it resembles the radial part of a Coulomb potential (as felt e.g. by an electron orbiting a nucleus). – Marek Jul 18 2011 at 16:01

The Cherednik algebra has a similar classification into discrete and unitary series: see arXiv:1106.5094 and arXiv:0901.4595. Strictly speaking, these papers classify the unitary irreducibles in category O. I don't know whether there is a larger category in which contravariant forms will exist, but anyway for the symmetric group category O will be closely tied to affine Lie algebras (thus to Virasoro) by the Arakawa-Suzuki functor, and to Hecke (thus TL algebras) by the Knizhnik-Zamolodchikov functor (which actually identifies O with the category of q-Schur modules for most values of the parameter). Maybe the Cherednik algebra can serve as a bridge between them: Etingof conjectures (true by case by case check for the symmetric group) that KZ of a unitary module is unitary, and it is true (again case by case) that via Arakawa-Suzuki the unitary modules (i.e. integrable modules) for affine $gl_n$ correspond to unitary modules for the Cherednik algebra.

At least for the symmetric group, the question of when there is a faithful unitary module in O is not very interesting: there is always one (either $L_c(triv)$ or $L_c(sign)$ will work). But if one is to make the connection to TL and the Virasoro algebra work probably one needs more detail.

Every Cherednik algebra module is in particular a module over a ring C[V] of polynomial functions on a vector space V, and its support is a subvariety of V. The faithful unitaries should be the unitaries with full support (I have not checked this, though one direction is obvious).

In the (much simpler) case of the Cherednik algebra of the symmetric group $S_n$, the algebra depends on one parameter c, which we may assume positive. The irreducibles in O are indexed by irreducible $S_n$-modules, and therefore by partitions of n. Writing $a(\lambda)$ for the largest hook length of the partition $\lambda$ and $b(\lambda)$ for a certain smaller hook length (see the paper of Etingof/Stoica for the precise def'ns), the corresponding irreducible $L_c(\lambda)$ is unitary iff $\lambda=(1^n)$ (corresponding to the sign representation), or $c \leq a(\lambda)$ or $c=1/m$ for a positive integer $m$ with $m \leq b(\lambda)$. The continuous part of the unitary set is precisely the closure of the set where the corresponding standard module is irreducible and unitary (this much is not surprising: the condition for the contravariant form to be positive definite on the standard module is open, and it's obviously pos. def. at $0$).

The module $L_c(\lambda)$ has full support iff: $c$ is not rational or $c=k/m$ and the partition is $m$-regular: the differences $\lambda_i-\lambda_{i+1}$ are strictly less than $m$. Thus $L_c(\lambda)$ is unitary of full support iff (1) $\lambda=(1^n)$, (2) $\lambda=(n)$ and $0 \leq c < 1/n$, (3) $\lambda \neq (n),(1^n)$ is a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with $m<b(\lambda)$, (4) $\lambda$ is not a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with $m \leq b(\lambda)$.

Taking the $n \rightarrow \infty$ limit of all this should be possible; I am running out of time again.

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 Very interesting. You say that "you get a picture very similar..." Could you please describe in a little bit more detail what you get for the Cherednik algebra associated to the infinite permutation group $S_\infty$? – André Henriques Jul 18 2011 at 1:56 I edited that bit to be more conservative; I don't know what the right definition of the Cherednik algebra of $S_\infty$ is, but I do think that one gets a meaningul definition of unitary set associated to a limiting shape of partitions''. – SPG Jul 18 2011 at 19:03 Don't know why the tex in the next to last paragraph is broken. Annoying. – SPG Jul 18 2011 at 19:13 I've taken the liberty of fixing the TeX in the second-last paragraph; I hope you don't mind. – Daniel Litt Jul 18 2011 at 19:52

I think it is not a coincidence, although the only relationship I can think of is a bit distant. Roughly, it goes: From positive energy representations of affine Kac-Moody algebras one gets certain values of $c$ in the discrete series. The TL algebras appear as centralizer algebras for quantum $\mathfrak{sl}_2$ (i.e. $End(V^{\otimes n})$ for $V$ the "vector" representation). At roots of unity one gets the TL discrete series (here $\delta$ is the $q$-dimension of $V$. On the other hand, the level-preserving tensor product on reps. of the affine Kac-Moody algebra of type $A$ gives a fusion category equivalent to the category one obtains from the quantum group situation (due to Finkelberg, although Lepowsky tells me there is a small gap that can be fixed using VOAs).

So I guess I am saying that there is a sort of Schur-Weyl duality relationship. This is not restricted to the type $A$ situation, for example, BMW algebras exhibit similar behavior which corresponds to the type $BCD$ quantum groups (or affine Kac-Moody algebras).

Probably the appropriate language to use is that of tensor categories associated with quantum groups. At roots of unity one gets unitary reps (see Wenzl or Xu's work on this) giving a discrete series which (at least combinatorially) corresponds to level-preserving fusion products for Kac-Moody algebras, which then are responsible for the Virasoro algebra situation. For non-roots of unity one still gets a continuous series of unitary reps.

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