14d. Toral conjectures

[math] \newcommand{\mathds}{\mathbb}[/math]

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Let us discuss now some further questions, in relation with the theory of toral subgroups, developed in chapter 13. We recall from there that associated to any closed subgroup [math]G\subset U_N^+[/math] is its diagonal torus, given by the following formula:

[[math]] C(T_1)=C(G)\Big/\left \lt u_{ij}=0\Big|\forall i\neq j\right \gt [[/math]]

More generally, given a closed subgroup [math]G\subset U_N^+[/math] and a matrix [math]Q\in U_N[/math], we let [math]T_Q\subset G[/math] be the diagonal torus of [math]G[/math], with fundamental representation spinned by [math]Q[/math]:

[[math]] C(T_Q)=C(G)\Big/\left \lt (QuQ^*)_{ij}=0\Big|\forall i\neq j\right \gt [[/math]]

This torus is then a group dual, given by the formula [math]T_Q=\widehat{\Lambda}_Q[/math], as usual up to the standard equivalence relation for the compact quantum groups, in order to avoid amenability issues, where [math]\Lambda_Q= \lt g_1,\ldots,g_N \gt [/math] is the discrete group generated by the following elements, which are unitaries inside the quotient algebra [math]C(T_Q)[/math]:

[[math]] g_i=(QuQ^*)_{ii} [[/math]]

As explained in chapter 13, the correct analogue of the maximal torus for [math]G\subset U_N^+[/math] is the collection of these spinned tori, called skeleton of [math]G[/math]:

[[math]] T=\left\{T_Q\subset G\big|Q\in U_N\right\} [[/math]]

Finally, let us recall from chapter 13 that several properties of [math]G[/math] are conjecturally encoded by the skeleton [math]T[/math], and with the conjectures being usually verified for the compact Lie groups, for the duals of the finitely generated discrete groups, and in a few other cases. We have the following result, from , complementing the material in chapter 13:

Theorem

The following hold, both over the category of compact Lie groups, and over the category of duals of finitely generated discrete groups:

• Characters: if [math]G[/math] is connected, for any nonzero [math]P\in C(G)_{central}[/math] there exists [math]Q\in U_N[/math] such that [math]P[/math] becomes nonzero, when mapped into [math]C(T_Q)[/math].
• Amenability: a closed subgroup [math]G\subset U_N^+[/math] is coamenable if and only if each of the tori [math]T_Q[/math] is coamenable, in the usual discrete group sense.
• Growth: assuming [math]G\subset U_N^+[/math], the discrete quantum group [math]\widehat{G}[/math] has polynomial growth if and only if each the discrete groups [math]\widehat{T_Q}[/math] has polynomial growth.

Show Proof

In the classical case, where [math]G\subset U_N[/math], the proof goes as follows:

(1) Characters. We can take here [math]Q\in U_N[/math] to be such that [math]QTQ^*\subset\mathbb T^N[/math], where [math]T\subset U_N[/math] is a maximal torus for [math]G[/math], and this gives the result.

(2) Amenability. This conjecture holds trivially in the classical case, [math]G\subset U_N[/math], due to the fact that these latter quantum groups are all coamenable.

(3) Growth. This is something nontrivial, well-known from the theory of compact Lie groups, and we refer here for instance to D'Andrea-Pinzari-Rossi .

Regarding now the group duals, here everything is trivial. Indeed, when the group duals are diagonally embedded we can take [math]Q=1[/math], and when the group duals are embedded by using a spinning matrix [math]Q\in U_N[/math], we can use precisely this matrix [math]Q[/math].

■

As in the previous chapter with the general results regarding the tori there, it is conjectured that the properties in Theorem 14.18 should hold in general. However, proving such things in general is probably something quite difficult, because Tannakian duality, which is basically our only tool, leads into fairly complicated combinatorial questions.

Following , as a first solid piece of evidence for the above conjectures, we have the following result, regarding the main examples of free quantum groups:

Theorem

The character, amenability and growth conjectures hold for the free quantum groups [math]G=O_N^+,U_N^+,S_N^+,H_N^+[/math].

Show Proof

We have [math]3\times4=12[/math] assertions to be proved, and the idea in each case will be that of using certain special group dual subgroups. We will mostly use the group dual subgroups coming at [math]Q=1[/math], which are well-known to be as follows:

[[math]] G=O_N^+,U_N^+,S_N^+,H_N^+\implies\Gamma_1=\mathbb Z_2^{*N},F_N,\{1\},\mathbb Z_2^{*N} [[/math]]

However, for some of our 12 questions, using these subgroups will not be enough, and we will use as well some carefully chosen subgroups of type [math]\Gamma_Q[/math], with [math]Q\neq1[/math].

As a last ingredient, we will need some specialized structure results for [math]G[/math], in the cases where [math]G[/math] is coamenable. Once again, the theory here is well-known, and the situations where [math]G=O_N^+,U_N^+,S_N^+,H_N^+[/math] is coamenable, along with the values of [math]G[/math], are as follows:

[[math]] \begin{cases} O_2^+=SU_2^{-1}\\ S_2^+=S_2,S_3^+=S_3,S_4^+=SO_3^{-1}\\ H_2^+=O_2^{-1} \end{cases} [[/math]]

To be more precise, the equalities [math]S_N^+=S_N[/math] at [math]N\leq3[/math] are known since Wang's paper , and the twisting results are all well-known, and we refer here to .

With these ingredients in hand, we can now go ahead with the proof. It is technically convenient to split the discussion over the 3 conjectures, as follows:

(1) Characters. For [math]G=O_N^+,U_N^+[/math], it is known that the algebra [math]C(G)_{central}[/math] is polynomial, respectively [math]*[/math]-polynomial, on the following variable:

[[math]] \chi=\sum_iu_{ii} [[/math]]

Thus, it is enough to show that the following variable generates a polynomial, respectively [math]*[/math]-polynomial algebra, inside the group algebra of [math]\mathbb Z_2^{*N},F_N[/math]:

[[math]] \rho=\sum_ig_i [[/math]]

But for the group [math]\mathbb Z_2^{*N}[/math] this is clear, and by using a multiplication by a unitary free from [math]\mathbb Z_2^{*N}[/math], the result holds as well for [math]F_N[/math].

Regarding now [math]G=S_N^+[/math], we have three cases to be discussed, as follows:

-- At [math]N=2,3[/math] this quantum group collapses to the usual permutation group [math]S_N[/math], and the character conjecture holds indeed.

-- At [math]N=4[/math] we have [math]S_4^+=SO_3^{-1}[/math], the fusion rules are the Clebsch-Gordan ones, and the algebra [math]C(G)_{central}[/math] is therefore polynomial on [math]\chi=\sum_iu_{ii}[/math]. Now observe that the spinned torus, with [math]Q=diag(F_2,F_2)[/math], is the following discrete group:

[[math]] \Gamma_Q=\mathbb Z_2*\mathbb Z_2=D_\infty [[/math]]

Since [math]Tr(u)=Tr(Q^*uQ)[/math], the image of [math]\chi=\sum_iu_{ii}[/math] in the quotient [math]C^*(\Gamma_Q)[/math] is the variable [math]\rho=2+g+h[/math], where [math]g,h[/math] are the generators of the two copies of [math]\mathbb Z_2[/math]. Now since this latter variable generates a polynomial algebra, we obtain the result.

-- At [math]N\geq5[/math] now, the fusion rules are once again the Clebsch-Gordan ones, the algebra [math]C(G)_{central}[/math] is, as before, polynomial on [math]\chi=\sum_iu_{ii}[/math], and the result follows by functoriality from the result at [math]N=4[/math], by using the embedding [math]S_4^+\subset S_N^+[/math].

Regarding now [math]G=H_N^+[/math], here it is known, from the computations in , that the algebra [math]C(G)_{central}[/math] is polynomial on the following two variables:

[[math]] \chi=\sum_iu_{ii}\quad,\quad \chi'=\sum_iu_{ii}^2 [[/math]]

We have two cases to be discussed, as follows:

-- At [math]N=2[/math] we have the following formula, which is well-known, and elementary:

[[math]] H_2^+=O_2^{-1} [[/math]]

Also, as explained in , with [math]Q=F_2[/math] we have:

[[math]] \Gamma_Q=D_\infty [[/math]]

Let us compute now the images [math]\rho,\rho'[/math] of the above variables [math]\chi,\chi'[/math] in the group algebra of [math]D_\infty[/math]. As before, from [math]Tr(u)=Tr(Q^*uQ)[/math] we obtain the following formula, where [math]g,h[/math] are the generators of the two copies of [math]\mathbb Z_2[/math]:

[[math]] \rho=g+h [[/math]]

Regarding now [math]\rho'[/math], let us first recall that the quotient map [math]C(H_2^+)\to C^*(D_\infty)[/math] is constructed as follows:

[[math]] \frac{1}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{pmatrix}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\to\begin{pmatrix}g&0\\0&h\end{pmatrix} [[/math]]

Equivalently, this quotient map is constructed as follows:

[[math]] \begin{eqnarray*} \begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{pmatrix} &\to&\frac{1}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\begin{pmatrix}g&0\\0&h\end{pmatrix}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\\ &=&\frac{1}{2}\begin{pmatrix}g+h&g-h\\g-h&g+h\end{pmatrix} \end{eqnarray*} [[/math]]

We can now compute the image of our character, as follows:

[[math]] \begin{eqnarray*} \rho' &=&\frac{1}{2}(g+h)^2\\ &=&\frac{1}{2}(2+2gh)\\ &=&1+gh \end{eqnarray*} [[/math]]

By using now the elementary fact that the variables [math]\rho=g+h[/math] and [math]\rho'=1+gh[/math] generate a polynomial algebra inside [math]C^*(D_\infty)[/math], this gives the result.

-- Finally, at [math]N\geq3[/math] the result follows by functoriality, via the standard diagonal inclusion [math]H_2^+\subset H_N^+[/math], from the result at [math]N=2[/math], that we established above.

(2) Amenability. Here the cases where [math]G[/math] is not coamenable are those of [math]O_N^+[/math] with [math]N\geq3[/math], [math]U_N^+[/math] with [math]N\geq2[/math], [math]S_N^+[/math] with [math]N\geq5[/math], and [math]H_N^+[/math] with [math]N\geq3[/math].

-- For [math]G=O_N^+,H_N^+[/math] with [math]N\geq3[/math] the result is clear, because the discrete group [math]\Gamma_1=\mathbb Z_2^{*N}[/math] is not amenable.

-- Clear as well is the result for [math]U_N^+[/math] with [math]N\geq2[/math], because the discrete group [math]\Gamma_1=F_N[/math] is not amenable either.

-- Finally, for [math]S_N^+[/math] with [math]N\geq5[/math] the result holds as well, because of the presence of Bichon's group dual subgroup [math]\widehat{\mathbb Z_2*\mathbb Z_3}[/math].

(3) Growth. Here the growth is polynomial precisely in the situations where [math]G[/math] is infinite and coamenable, the precise cases being:

[[math]] O_2^+=SU_2^{-1}\quad,\quad S_4^+=SO_3^{-1}\quad,\quad H_2^+=O_2^{-1} [[/math]]

With these formulae in hand, the result follows from the well-known fact that the growth invariants are stable under twisting.

■

With a bit more work, the above result from can be extended to general quantum reflection groups [math]H_N^{s+}[/math] as well, and in particular to the quantum group [math]K_N^+[/math], and we conclude that our conjectures hold for the main easy quantum groups, namely:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

As a second piece of evidence now for our conjectures, of different nature, we will prove that these conjectures hold for any half-classical quantum group. In order to do so, we can use the modern approach to half-liberation, from Bichon and Dubois-Violette , based on crossed products and related [math]2\times2[/math] matrix models, as follows:

Theorem

Given a conjugation-stable closed subgroup [math]H\subset U_N[/math], consider the algebra [math]C([H])\subset M_2(C(H))[/math] generated by the following variables:

[[math]] u_{ij}=\begin{pmatrix}0&v_{ij}\\ \bar{v}_{ij}&0\end{pmatrix} [[/math]]

Then [math][H][/math] is a compact quantum group, we have [math][H]\subset O_N^*[/math], and any non-classical subgroup [math]G\subset O_N^*[/math] appears in this way, with [math]G=O_N^*[/math] itself appearing from [math]H=U_N[/math].

Show Proof

We have several things to be proved, the idea being as follows:

(1) As a first observation, the matrices in the statement are self-adjoint. Let us prove now that these matrices are orthogonal. We have:

[[math]] \begin{eqnarray*} \sum_ku_{ik}u_{jk} &=&\sum_k\begin{pmatrix}0&v_{ik}\\ \bar{v}_{ik}&0\end{pmatrix} \begin{pmatrix}0&v_{jk}\\ \bar{v}_{jk}&0\end{pmatrix}\\ &=&\sum_k\begin{pmatrix}v_{ik}\bar{v}_{jk}&0\\ 0&\bar{v}_{ik}v_{jk}\end{pmatrix}\\ &=&\begin{pmatrix}1&0\\0&1\end{pmatrix} \end{eqnarray*} [[/math]]

In the other sense, the computation is similar, as follows:

[[math]] \begin{eqnarray*} \sum_ku_{ki}u_{kj} &=&\sum_k\begin{pmatrix}0&v_{ki}\\ \bar{v}_{ki}&0\end{pmatrix} \begin{pmatrix}0&v_{kj}\\ \bar{v}_{kj}&0\end{pmatrix}\\ &=&\sum_k\begin{pmatrix}v_{ki}\bar{v}_{kj}&0\\ 0&\bar{v}_{ki}v_{kj}\end{pmatrix}\\ &=&\begin{pmatrix}1&0\\0&1\end{pmatrix} \end{eqnarray*} [[/math]]

(2) Our second claim is that the matrices in the statement half-commute. Consider indeed arbitrary antidiagonal [math]2\times2[/math] matrices, with commuting entries, as follows:

[[math]] X_i=\begin{pmatrix}0&x_i\\ y_i&0\end{pmatrix} [[/math]]

We have then the following computation:

[[math]] \begin{eqnarray*} X_iX_jX_k &=&\begin{pmatrix}0&x_i\\ y_i&0\end{pmatrix}\begin{pmatrix}0&x_j\\ y_j&0\end{pmatrix}\begin{pmatrix}0&x_k\\ y_k&0\end{pmatrix}\\ &=&\begin{pmatrix}0&x_iy_jx_k\\ y_ix_jy_k&0\end{pmatrix} \end{eqnarray*} [[/math]]

Since this quantity is symmetric in [math]i,k[/math], we obtain, as desired:

[[math]] X_iX_jX_k=X_kX_jX_i [[/math]]

(3) According now to the definition of the quantum group [math]O_N^*[/math], we have a representation of algebras, as follows where [math]w[/math] is the fundamental corepresentation of [math]C(O_N^*)[/math]:

[[math]] \pi:C(O_N^*)\to M_2(C(H))\quad,\quad w_{ij}\to u_{ij} [[/math]]

Thus, with the compact quantum space [math][H][/math] being constructed as in the statement, we have a representation of algebras, as follows:

[[math]] \rho:C(O_N^*)\to C([H])\quad,\quad w_{ij}\to u_{ij} [[/math]]

(4) With this in hand, it is routine to check that the compact quantum space [math][H][/math] constructed in the statement is indeed a compact quantum group, with this being best viewed via an equivalent construction, with a quantum group embedding as follows:

[[math]] C([H])\subset C(H)\rtimes\mathbb Z_2 [[/math]]

(5) As for the proof of the converse, stating that any non-classical subgroup [math]G\subset O_N^*[/math] appears in this way, this is something more tricky, and we refer here to .

(6) Finally, for the fact that we have indeed [math]O_N^*=[U_N][/math], we refer here as well to . We will be back to this as well in chapter 16 below, with a direct analytic proof of this, based on the fact that the representation [math]\rho[/math] constructed above, with [math]H=U_N[/math], commutes with the respective Haar functionals, and so must be faithful.

■

In relation with the above, we will need as well the following result, regarding the irreducible corepresentations, also from Bichon-Dubois-Violette :

Theorem

In the context of the correspondence [math]H\to[H][/math] we have a bijection

[[math]] Irr([H])\simeq Irr_0(H)\coprod Irr_1(H) [[/math]]

where the sets on the right are given by

[[math]] Irr_k(H)=\left\{r\in Irr(H)\Big|\exists l\in\mathbb N,r\in u^{\otimes k}\otimes(u\otimes\bar{u})^{\otimes l}\right\} [[/math]]

induced by the canonical identification [math]Irr(H\rtimes\mathbb Z_2)\simeq Irr(H)\coprod Irr(H)[/math].

Show Proof

This is something more technical, also from . It is easy to see that we have an equality of projective versions [math]P[H]=PH[/math], which gives an inclusion as follows:

[[math]] Irr_0(H)=Irr(PH)\subset Irr([H]) [[/math]]

As for the remaining irreducible representations of [math][H][/math], these must come from an inclusion [math]Irr_1(H)\subset Irr([H])[/math], appearing as above. See .

■

Now back the maximal tori, the situation here is very simple, as follows:

Proposition

The group dual subgroups [math]\widehat{[\Gamma]}_Q\subset[H][/math] appear via

[[math]] [\Gamma]_Q=[\Gamma_Q] [[/math]]

from the group dual subgroups [math]\widehat{\Gamma}_Q\subset H[/math] associated to [math]H\subset U_N[/math].

Show Proof

Let us first discuss the case [math]Q=1[/math]. Consider the diagonal subgroup [math]\widehat{\Gamma}_1\subset H[/math], with the associated quotient map [math]C(H)\to C(\widehat{\Gamma}_1)[/math] denoted:

[[math]] v_{ij}\to\delta_{ij}h_i [[/math]]

At the level of the algebras of [math]2\times2[/math] matrices, this map induces a quotient map:

[[math]] M_2(C(H))\to M_2(C(\widehat{\Gamma}_1)) [[/math]]

Our claim is that we have a factorization, as follows:

[[math]] \begin{matrix} C([H])&\subset&M_2(C(H))\\ \\ \downarrow&&\downarrow\\ \\ C([\widehat{\Gamma}_1])&\subset&M_2(C(\widehat{\Gamma}_1)) \end{matrix} [[/math]]

Indeed, it is enough to show that the standard generators of [math]C([H])[/math] and of [math] C([\widehat{\Gamma}_1])[/math] map to the same elements of [math]M_2(C(\widehat{\Gamma}_1))[/math]. But these generators map indeed as follows:

[[math]] \begin{matrix} u_{ij}&\to&\begin{pmatrix}0&v_{ij}\\ \bar{v}_{ij}&0\end{pmatrix}\\ \\ &&\downarrow\\ \\ \delta_{ij}v_{ij}&\to&\begin{pmatrix}0&\delta_{ij}h_i\\ \delta_{ij}h_i^{-1}&0\end{pmatrix} \end{matrix} [[/math]]

Thus we have the above factorization, and since the map on the left is obtained by imposing the relations [math]u_{ij}=0[/math] with [math]i\neq j[/math], we obtain, as desired:

[[math]] [\Gamma]_1=[\Gamma_1] [[/math]]

In the general case now, [math]Q\in U_N[/math], the result follows by applying the above [math]Q=1[/math] result to the quantum group [math][H][/math], with fundamental corepresentation [math]w=QuQ^*[/math].

■

Now back to our conjectures, we have the following result, from :

Theorem

The [math]3[/math] toral conjectures, regarding the characters, amenability and growth, hold for any half-classical quantum group of the form

[[math]] [H]\subset O_N^* [[/math]]

with [math]H\subset U_N[/math] being connected.

Show Proof

We know that the conjectures hold for [math]H\subset U_N[/math]. The idea will be that of “transporting” these results, via [math]H\to [H][/math]:

(1) Characters. We can pick here a maximal torus [math]T=\Gamma_Q[/math] for the compact group [math]H\subset U_N[/math], and by using the formula [math][\Gamma]_Q=[\Gamma_Q]=[T][/math] from Proposition 14.21 above, we obtain the result, via the identification in Theorem 14.20.

(2) Amenability. There is nothing to be proved here, because [math]O_N^*[/math] is coamenable, and so are all its quantum subgroups. Note however, in relation with the various comments made in chapter 3 above, that in the connected case, the Kesten measures of [math]G,[T][/math] are intimately related. For some explicit formulae here, for [math]G=O_N^*[/math] itself, see .

(3) Growth. Here the situation is similar to the one for the amenability conjecture, because the quantum group [math][H][/math] has polynomial growth.

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Let us mention that the above results can be extended to the general, unitary half-classical case, by using some suitable variations of the [math]2\times2[/math] matrix models used in the above. We refer here to and related papers, for the most dealing with noncommutative geometry in general, where such questions were investigated.

As a conclusion now, the theory of maximal tori developed in this chapter and in the previous one looks like something quite promising, worth some further investigation. Unfortunately, and as usual in mathematics and physics with new topics, things going on slowly here, with the various communities hesitating of getting into the subject, and with the unaffiliated individuals being a dying breed, in these modern times.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].