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COMPACT QUANTUM GROUPS GENERATED BY THEIR TORI
Teo Banica
To cite this version:
Teo Banica. COMPACT QUANTUM GROUPS GENERATED BY THEIR TORI. 2019. �hal- 02098565�
TEO BANICA
Abstract. Associated to any closed subgroupG⊂UN+ is a family of toral subgroups TQ⊂G, indexed by the unitary matricesQ∈UN. The family{TQ|Q∈UN}is expected to encode the main properties ofG, and there are several conjectures in this sense. We verify here the generation conjecture,G=< TQ|Q∈UN >, for various classes of compact quantum groups. Our results generalize the previously known facts on the subject.
CONTENTS
Introduction 1
1. Toral subgroups 2
2. The conjectures 4
3. Toral generation 6
4. Stability results 8
5. Group duals 10
6. Quantum permutations 11
7. Liberation theory 13
8. Open problems 14
References 15
Introduction
The structure and classification of the closed subgroups G ⊂ UN+ is largely unknown.
Among the ideas which appeared in recent years is the fact that the properties ofGshould be encoded by a certain family of group duals TQ ⊂ G, indexed by the unitary matrices Q∈UN, which altogether should play the role of “maximal torus” for G.
The definition of these group duals TQ ⊂Gis very simple, as follows:
C(TQ) =C(G)/ <(QuQ∗)ij = 0,∀i6=j >
Indeed, we obtain in this way a Hopf algebra, and since the generators gi = (QuQ∗)ii are group-like in the quotient, ∆(gi) =gi⊗gi, we have a group algebra.
2010Mathematics Subject Classification. 46L65 (22C05).
Key words and phrases. Compact quantum group, Toral subgroup.
1
Following the nonocommutative geometry philosophy, where group duals can be thought of as being “tori”, these group duals TQ ⊂G are called “standard tori” of G.
All this goes back to [10], and then to [4], in connection with representation theory and noncommutative geometry questions. In a more recent paper [8], the family{TQ|Q∈UN} was shown to encode the main properties of G, in many interesting cases, and 3 general conjectures were formulated, regarding characters, amenability and growth.
A fourth, and even more basic conjecture on the subject, is the generation one from [1], which simply states that Gshould be generated by its tori:
G=< TQ|Q∈UN >
The paper [1] was in fact a survey, and when attempting to survey [4], [8], [10], we run into this simple and beautiful statement. So, instead of surveying [4], [8], [10], we presented there this new conjecture, with the promise to come back later to it.
In this paper we are back to this. Our main result is as follows:
Theorem. The free quantum groups G⊂UN+ are generated by their tori.
The proof uses the elementary fact that the property of being generated by its tori holds for the classical groups, and is stable as well under topological generation < G, H >, then the standard formula G =< Gclass, SN+ > from [9], the more advanced formula SN+ =< SN, SN−1+ > atN ≥5, from [14], and an argument based on [5] forS4+.
The paper is organized as follows: 1-2 are preliminary sections, in 3-4 we discuss in detail the generation conjecture, in 5-6 we gather some technical results, and in 7-8 we present our main results, and we end with some concluding remarks.
1. Toral subgroups
We use the compact matrix quantum group formalism developed by Woronowicz in [27], [28], with the extra axiom that the square of the antipode is the identity, S2 =id.
With this extra axiom imposed, the quantum groups in [27], [28] coincide with the closed quantum subgroups G⊂UN+ of Wang’s free unitary quantum group [25].
We are interested here in group dual subgroups. Let us begin with:
Proposition 1.1. Let G⊂UN+ be a compact quantum group, and consider the group dual subgroups Λb ⊂G, also called toral subgroups, or simply “tori”.
(1) In the classical case, where G ⊂ UN is a compact Lie group, these are the usual tori, where by torus we mean here closed abelian subgroup.
(2) In the group dual case, G = Γb with Γ =< g1, . . . , gN > being a discrete group, these are the duals of the various quotientsΓ→Λ.
Proof. Both these assertions are elementary, as follows:
(1) This follows indeed from the fact that a closed subgroup T ⊂ UN+ is at the same time classical, and a group dual, precisely when it is classical and abelian.
(2) This follows from the general propreties of the Pontrjagin duality, and more precisely from the fact that the subgroups Λb ⊂bΓ correspond to the quotients Γ→Λ.
There are two motivations for the study of such subgroups. First, it is well-known that the fine structure of a compact Lie groupG⊂UN is encoded by its maximal torus. Thus, we can expect the tori to encode interesting information, in general. See [8], [10].
As a second motivation, any action GyX will produce actions Λb yX. And, due to the fact that Λ are very familiar objects, namely discrete groups, these latter actions are easier to study, and can ultimately lead to results aboutGyX itself. See [4].
We have the following key construction, from [4], [8], [10]:
Proposition 1.2. Given a subgroup G⊂UN+ and a matrix Q∈UN, the construction C(TQ) =C(G). D
(QuQ∗)ij = 0
∀i6=jE
produces a group dual,TQ =ΛbQ, whereΛQ=< g1, . . . , gN >is the discrete group generated by the elements gi = (QuQ∗)ii, which are unitaries inside C(TQ).
Proof. Since the matrix v = QuQ∗ is a unitary corepresentation, its diagonal entries gi =vii, when regarded inside C(TQ), are unitaries, and satisfy:
∆(gi) =gi⊗gi
Thus the quotient algebra C(TQ) is a group algebra, and more specifically we have C(TQ) =C∗(ΛQ), where ΛQ =< g1, . . . , gN >is the group in the statement,.
Summarizing, associated to any closed subgroup G ⊂ UN+ is a whole family of tori, indexed by the unitaries U ∈UN. We use the following terminology:
Definition 1.3. Let G⊂UN+ be a closed subgroup.
(1) The tori TQ⊂G constructed above are called standard tori of G.
(2) The collection of toriT = TQ
Q∈UN is called skeleton of G.
The terminology here is new, upgrading some previous terminologies from [1], [4], [8], all suffering from minor flaws. The difficulty in naming things comes from the fact that the family T ={TQ|Q∈UN} plays the role of a “maximal torus” of G.
Let us first discuss the examples. In the classical case, the result is as follows:
Proposition 1.4. For a closed subgroup G⊂UN we have TQ =G∩(Q∗TNQ) where TN ⊂UN is the group of diagonal unitary matrices.
Proof. This is indeed clear at Q = 1, where Γ1 appears by definition as the dual of the compact abelian group G∩TN. In general, this follows by conjugating by Q.
In the group dual case we have the following result, from [4]:
Proposition 1.5. Given a discrete group Γ =< g1, . . . , gN >, consider its dual compact quantum group G=bΓ, diagonally embedded into UN+. We have then
ΛQ= Γ/ < gi =gj|∃k, Qki 6= 0, Qkj 6= 0>
with the embedding TQ ⊂G=bΓ coming from the quotient map Γ→ΛQ.
Proof. Assume indeed that Γ =< g1, . . . , gN > is a discrete group, withΓb ⊂ UN+ coming via u=diag(g1, . . . , gN). With v =QuQ∗, we have:
X
s
Q¯sivsk =X
st
Q¯siQstQ¯ktgt =X
t
δitQ¯ktgt = ¯Qkigi
Thus vij = 0 for i 6=j gives ¯Qkivkk = ¯Qkigi, which is the same as saying that Qki 6= 0 implies gi = vkk. But this latter equality reads gi = P
j|Qkj|2gj, and we conclude that Qki 6= 0, Qkj 6= 0 impliesgi =gj, as desired. The converse holds too. See [4].
As a first general result now, we have the following well-known fact:
Theorem 1.6. Any torus T ⊂G appears as follows, for a certain Q∈UN: T ⊂TQ ⊂G
In other words, any torus appears inside a standard torus.
Proof. It is known from [27] that each torus T =Λb ⊂UN+, coming from a discrete group Λ =< g1, . . . , gN >, has a fundamental corepresentation as follows, with Q∈UN:
u=Qdiag(g1, . . . , gN)Q∗
But this shows that we have T ⊂TQ, and this gives the result.
We refer to [4], [8] for more details regarding the above material.
2. The conjectures
According to the above results, we can expect the skeletonT to encode various algebraic and analytic properties ofG. We survey here the various conjectures on the subject.
As a first statement, coming from an observation from [1], we have:
Conjecture 2.1 (Injectivity). The construction G → T is injective, in the sense that G6=H implies TQ(G)6=TQ(H), for some Q∈UN.
Here, and in what follows, the closed subgroups G ⊂ UN+ are by definition identified in the case where the corresponding dense∗-algebras of coefficients are isomorphic, via a
∗-morphism mapping standard coordinates to standard coordinates.
The above conjecture is of course something quite abstract, and probably quite out of reach, for the moment. Here is a softer version of this conjecture, also from [1]:
Conjecture 2.2 (Monotony). The construction G → T is increasing, in the sense that passing to a subgroup H ⊂G decreases at least one of the tori TQ.
Once again, what we have here looks like an abstract, difficult statement. In addition, such kind of statement is not really in tune with what we want to do, namely expressing the properties of Gin terms of those of T, which is something very concrete.
Here is now the main conjecture from [1], which definitely looks better:
Conjecture 2.3 (Generation). Any closed quantum subgroup G ⊂ UN+ has the property G=< TQ|Q∈UN >. In other words, G is generated by its tori.
Here the generation operation< , > is taken of course in a topological sense. We refer to [16], where this operation was introduced, or to the survey paper [7]. We will be back with full details here, this conjecture being the one that we are interested in.
Observe that we can indeed replace in the final conclusion the standard tori by the arbitrary tori, as it was done above, and this due to Theorem 1.6.
Let us review now as well the original conjectures from [8], which are much more concrete, but perhaps less fundamental statements. First, we have:
Conjecture 2.4 (Characters). If G is connected, for any nonzero P ∈ C(G)central there exists Q∈UN such that P becomes nonzero, when mapped into C(TQ).
Here the connectivity assumption states that there is no finite quantum group quotient G → F 6= {1}. This is equivalent to the fact that the coefficient algebra < rij > must be infinite dimensional, for any nontrivial irreducible unitary representation r. For the group duals, G=Γ, this is the same as asking for Γ to have no torsion. See [8].b
Here is a related conjecture, which probably looks a bit better:
Conjecture 2.5 (Amenability). A closed subgroup G⊂UN+ is coamenable if and only if each of the tori TQ is coamenable.
Observe that in one sense this is trivial, because the quotient map C(G)→C(TQ) can be interpreted as coming from a discrete quantum group quotient map Gb →TcQ.
In the other sense, an equivalent statement, perhaps a bit more convenient, is that if Gb is not amenable, then there exists Q∈UN such thatTcQ is not amenable.
Finally, here is a more specialized statement, regarding the growth:
Conjecture 2.6 (Growth). Assuming G ⊂UN+, the discrete quantum group Gb has poly- nomial growth if and only if each TcQ has polynomial growth.
This conjecture is actually only the “tip of the iceberg”, and there as well a series of more specialized conjectures, regarding the cardinality, the polynomial growth exponents, and the exponential growth exponents, based on the work in [17], [18]. See [8].
As a first general statement now, coming from [1], [8], we have:
Theorem 2.7. The above 6conjectures, involving the standard tori, hold for the classical groups, and for the group duals as well.
Proof. In the classical case, whereG⊂UN, the proof goes as follows:
(1) Injectivity. This follows from the generation conjecture, explained below.
(2) Monotony. Once again, this follows from the generation conjecture.
(3) Generation. We use the formula TQ = G∩Q∗TNQ, from Proposition 1.4. Since any group element U ∈G is diagonalizable, U =Q∗DQ with Q∈UN, D ∈TN, we have U ∈TQ for this value ofQ∈UN, and this gives the result. See [1].
(4) Characters. We can take here Q∈UN to be such thatQT Q∗ ⊂TN, where T ⊂UN is a maximal torus for G, and this gives the result. See [8].
(5) Amenability. This conjecture holds trivially in the classical case, G ⊂UN, due to the fact that these latter quantum groups are all coamenable. See [8].
(6) Growth. This is something nontrivial, and we refer here to [18]. The above- mentioned more specialized estimates follow as well from [18]. See [8].
Regarding now the group duals, here everything is trivial. Indeed, when the group duals are diagonally embedded we can takeQ= 1, and when the group duals are embedded by using a spinning matrix Q∈UN, we can use precisely this matrix Q. See [1], [8].
3. Toral generation
In what follows we focus on the generation conjecture. Let us begin with a more detailed formulation of what we have already, as follows:
Proposition 3.1. The generation conjecture G=< TQ|Q∈UN > holds, due to
G= [
Q∈UN
TQ
both for the compact Lie groups, G⊂UN, and for the group duals, G=bΓ.
Proof. This follows from the proof of Theorem 2.7. Indeed, in the compact Lie group case, G ⊂ UN, any group element U ∈ G must belong to one of the tori TQ, and this gives the result. As for the group dual case, where G=Γ is embedded intob UN+ by using a spinning matrix Q∈UN, here we have by definitionG=TQ, and we are done.
In general, in order to approach the generation conjecture, we can use the Tannakian approach to the compact quantum groups, coming from [28], as follows:
Proposition 3.2. The closed subgroups G ⊂ UN+ are in correspondence with their Tan- nakian categories C(k, l) =Hom(u⊗k, u⊗l), the correspondence being given by
C(G) = C(UN+). D
T ∈Hom(u⊗k, u⊗l)
∀k, l,∀T ∈C(k, l)E
where all the exponents are by definition colored integers, with the corresponding tensor powers being defined by u⊗∅ = 1, u⊗◦ =u, u⊗• = ¯u and multiplicativity.
Proof. The idea indeed is that we have a surjective arrow from left to right, and the injectivity comes either from [28], or can be checked directly, by doing some algebra, and then by applying the bicommutant theorem, as a main tool. See [21].
With the above result in hand, we can formulate:
Proposition 3.3. The closed subgroups of UN+ are subject to ∩ and < , > operations, constucted via the above Tannakian correspondence G→CG, as follows:
(1) Intersection: defined via CG∩H =< CG, CH >.
(2) Generation: defined via C<G,H>=CG∩CH.
In the classical case, where G, H ⊂UN, we obtain in this way the usual notions.
Proof. Since the ∩ and < , > operations are clearly well-defined for the Tannakian cat- egories, the operations in (1,2) make sense indeed. As for the last assertion, this is something well-known, which follows from an elementary computation. See [16].
The above statement is of course something quite compact. For further details on all this, we refer to the original paper [16], or to the recent survey article [7].
In relation now with the present questions, we first have:
Proposition 3.4. Given a closed subgroup G ⊂ UN+ and a matrix Q ∈ UN, the corre- sponding standard torus and its Tannakian category are given by
TQ =G∩TQ , CTQ =< CG, CTQ >
where TQ ⊂ UN+ is the dual of the free group FN =< g1, . . . , gN >, with the fundamental corepresentation of C(TQ) being the matrix u=Qdiag(g1, . . . , gN)Q∗.
Proof. The first assertion comes from the well-known fact that given two closed subgroups G, H ⊂UN+, the corresponding quotient algebra C(UN+)→C(G∩H) appears by dividing by the kernels of both the quotient maps C(UN+)→C(G) and C(UN+)→C(H).
Indeed, the construction of TQ from Proposition 1.3 amounts precisely in performing this operation, with H =TQ, and so we obtain TQ =G∩TQ, as claimed.
As for the Tannakian category formula, this follows from this, and from the general duality formula CG∩H =< CG, CH >from Proposition 3.3 above.
Regarding now the toral generation, we have the following result:
Theorem 3.5. Given a closed subgroup G ⊂ UN+, the subgroup G0 =< TQ|Q ∈ UN >
generated by its standard tori has the following Tannakian category:
CG0 = \
Q∈UN
< CG, CTQ >
In particular we have G=G0 when this intersection reduces to CG.
Proof. Consider indeed the subgroup G0 ⊂Gconstructed in the statement. According to the general formula C<G,H>=CG∩CH from Proposition 3.3, we have:
CG0 = \
Q∈UN
CTQ
Together with the formula in Proposition 3.4, this gives the result.
Summarizing, the toral generation conjecture admits a combinatorial formulation, as above. We should mention that some related considerations already appeared in [8], and then in [1], but with much more obscure formulations. The above simple statement is a corollary of Proposition 3.3, whose present formulation is something quite recent.
4. Stability results
In this section we discuss various stability properties of the generation conjecture.
Let us first discuss the stability under topological generation. We first have:
Proposition 4.1. Given two closed subgroups G, H ⊂UN+, we have
< TQ(G), TQ(H)>⊂TQ(< G, H >) for any unitary matrix Q∈UN.
Proof. This can be proved either by using Proposition 3.4, or directly. For the direct proof, which is perhaps the simplest, with notations from section 3 above, we have:
TQ(G) = G∩TQ ⊂< G, H >∩TQ =TQ(< G, H >) TQ(H) = H∩TQ ⊂< G, H >∩TQ =TQ(< G, H >)
Now since A, B ⊂C implies< A, B >⊂C, this gives the result.
Regarding now the generation conjecture, we have here:
Proposition 4.2. The generation conjecture is stable under < , >.
Proof. Assuming that two closed subgroups G, H ⊂UN+ are both generated by their tori, we have the following computation, using the inclusions in Proposition 4.1:
< G, H > = << TQ(G)|Q∈UN >, < TQ(H)|Q∈UN >>
= < TQ(G), TQ(H)|Q∈UN >
= << TQ(G), TQ(H)>|Q∈UN >
⊂ < TQ(< G, H >)|Q∈UN >
Thus the quantum group < G, H >is generated by its tori, as claimed.
In relation now with the intersection operation, we have:
Proposition 4.3. Given two closed subgroups G, H ⊂UN+, we have TQ(G∩H) = TQ(G)∩TQ(H)
for any unitary matrix Q∈UN.
Proof. We have indeed the following computation:
TQ(G∩H) = (G∩H)∩TQ
= (G∩TQ)∩(H∩TQ)
= TQ(G)∩TQ(H)
This result was actually already pointed out in [1], with a longer proof.
Observe that the stability of the generation conjecture under intersections does not follow from this. Indeed, assuming that G, H ⊂UN+ are both generated by their tori, the only computation that we can do is as follows, leading to a triviality:
< TQ(G∩H)|Q∈UN > = < TQ(G)∩TQ(H)|Q∈UN >
⊂ < TQ(G)|Q∈UN >∩< TQ(H)|Q∈UN >
= G∩H
Thus, the intersection question is probably a quite difficult one.
Let us discuss now product operations. We first have here:
Proposition 4.4. We have the following formula, for any G, H and R, S, TR⊗S(G×H) = TR(G)×TS(H)
and the generation conjecture is stable under usual products ×.
Proof. The formula in the statement is clear from definitions. By using this formula, when assuming that both G, H are generated by their tori, we have:
< TQ(G×H)|Q∈UM N > ⊃ < TR⊗S(G×H)|R∈UM, S ∈UN >
= < TR(G)×TS(H)|R ∈UM, S ∈UN >
= < TR(G)× {1},{1} ×TS(H)|R∈UM, S ∈UN >
= < TR(G)|R ∈UM >×< TG(H)|H ∈UN >
= G×H
Thus,G×H is generated as well by its tori, as claimed.
For the dual free products the situation is nearly identical, as follows:
Proposition 4.5. We have the following formula, for any G, H and R, S, TR⊗S(Gˆ∗H) = TR(G) ˆ∗TS(H)
and the generation conjecture is stable under dual free products ˆ∗.
Proof. Once again, the formula in the statement is clear. By using this formula, when assuming that both G, H are generated by their tori, we have:
< TQ(Gˆ∗H)|Q∈UM N > ⊃ < TR⊗S(Gˆ∗H)|R ∈UM, S∈UN >
= < TR(G) ˆ∗TS(H)|R∈UM, S ∈UN >
= < TR(G)× {1},{1} ×TS(H)|R ∈UM, S∈UN >
= < TR(G)|R∈UM > ˆ∗ < TG(H)|H ∈UN >
= Gˆ∗H
Thus,Gˆ∗H is generated as well by its tori, as claimed.
As a conclusion to these considerations, we have:
Theorem 4.6. The generation conjecture is stable under:
(1) Generations < G, H >.
(2) Usual products G×H.
(3) Dual free products G∗ˆH.
Proof. This follows indeed from the above results.
We will discuss as well a number of more subtle operations, later on.
5. Group duals
In order to advance, and to prove some technical results as well, we will need full information regarding the group duals which appear in section 3 above.
Regarding the intersection operation, the result here is as follows:
Proposition 5.1. The intersection of diagonally embedded group duals is given by bΓ∩Λ =b Θb
with the quotientFN →Θbeing obtained by dividing the free groupFN by both the kernels of the quotient maps FN →Γ and FN →Λ.
Proof. This comes from the general properties of the intersection operation ∩, and more specifically from the fact thatC(UN+)→C(G∩H) is obtained by dividingC(UN+) by both the kernels of the quotient maps C(UN+)→C(G) andC(UN+)→C(H). See [16].
In order to discuss now the generation operation, we will need:
Proposition 5.2. The Tannakian category for a diagonally embedded group dual bΓ con- sists of the linear maps T satisfying the condition
gj1. . . gjk 6=gi1. . . gil =⇒ Ti1...il,j1...jk = 0 with exponents added, in the colored case.
Proof. We have the following computation:
T u⊗k = X
ij
ei1...il,j1...jkTi1...il,j1...jkX
j
ej1...jk,j1...jk ⊗gj1. . . gjk
= X
ij
ei1...il,j1...jkTi1...il,j1...jk ⊗gj1. . . gjk
We have as well the following computation:
u⊗lT = X
i
ei1...il,i1...il⊗gi1. . . gilX
ij
ei1...il,j1...jkTi1...il,j1...jk
= X
ij
ei1...il,j1...jkTi1...il,j1...jk⊗gi1. . . gil
But this proves our claim, and in the colored case the proof is similar.
Observe that the above result agrees with the Tannakian definition of the intersection.
To be more precise, the Tannakian category of group bΓ∩Λ from Proposition 5.1 aboveb appears indeed as the category generated by the Tannakian categories of bΓ,Λ.b
With the above results in hand, we can now prove:
Theorem 5.3. The generation operation for diagonally embedded group duals is given by D
bΓ,ΛbE
=Θb
with the quotient FN →Θbeing obtained by dividing the free group FN by the intersection of the kernels of the quotient maps FN →Γ and FN →Λ.
Proof. This follows from Proposition 5.2 above, via a number of standard identifications.
To be more precise, we know from there that the generation relations are:
gi1. . . gik =gj1. . . gjl , hi1. . . hik =hj1. . . hjl =⇒ ti1. . . tik =tj1. . . tjl
But this gives the formula in the statement, via Frobenius duality.
Summarizing, our various notions and operations correspond to very basic notions and operations from discrete group theory.
6. Quantum permutations
We discuss here the generation conjecture for the quantum permutation groups, con- structed in [26]. Let us first recall from [11] that we have the following result:
Proposition 6.1. The group dual subgroups Γb⊂SN+ appear from the quotients ZN1 ∗. . .∗ZNk →Γ
with N1+. . .+Nk =N, regarded as a partition of N.
Proof. The fact that each quotient as in the statement produces indeed a quantum per- mutation group is standard, and can be checked as follows:
bΓ ⊂ ZN1 ∗\. . .∗ZNk =ZdN1∗ˆ. . .∗ ˆ∗ZdNk
' ZN1ˆ∗ . . .∗ ˆ∗ZNk ⊂SN1∗ˆ. . .∗ ˆ∗SNk
⊂ SN+
1∗ˆ. . .∗ ˆ∗SN+
k ⊂SN+
1+...+Nk
As for the converse, this follows from the orbit theory developed in [11].
We have as well the following related result, from [4]:
Proposition 6.2. For the quantum permutation group G=SN+, we have:
(1) Given Q∈UN, the quotient FN →ΛQ comes from the following relations:
gi = 1 if P
lQil 6= 0 gigj = 1 if P
lQilQjl6= 0 gigjgk = 1 if P
lQilQjlQkl 6= 0
(2) Given a decompositionN =N1+. . .+Nk, for the matrix Q=diag(FN1, . . . , FNk), where FN = √1
N(ξij)ij with ξ =e2πi/N is the Fourier matrix, we obtain:
ΛQ =ZN1 ∗. . .∗ZNk
(3) Given an arbitrary matrixQ∈UN, there exists a decompositionN =N1+. . .+Nk, such that ΛQ appears as quotient of ZN1 ∗. . .∗ZNk.
Proof. All this is standard, and the final conclusion itself is something that we already
know, coming from Proposition 6.1 above. See [4].
Let us discuss now the generation conjecture. We first have:
Proposition 6.3. The generation conjecture for SN+ with N ≥4 is equivalent to
\
N=N1+...+Nk
CZN
1ˆ∗...ˆ∗ZNk =span(N C)
where on the right we have the span of the category of noncrossing partitions.
Proof. This follows indeed from the above results. To be more precise, this simply follows from the general results from section 3, and from Proposition 6.1 above.
AtN = 4 now, we have the following result:
Proposition 6.4. We have the generation formula S4+=< S4,Dd∞ >
and so the generation conjecture holds for S4+.
Proof. This is our key result, and there are several proofs for it, as follows:
(1) A first method, which is purely combinatorial, is via considerations similar to those in Proposition 6.3 above. Indeed, according to Proposition 6.2, the unitaryQ∈U4 which produces the group dual Dd∞ as a corresponding standard torus is as follows:
Q= 1
√2
1 1 0 0
1 −1 0 0
0 0 1 1
0 0 1 −1
With this choice of Q, the formula that we want to prove is as follows:
span(P)∩CTQ =span(N C)
But this can be done with some combinatorics, not to be detailed here.
(2) A second method, which is more conceptual, is by using the standard identification S4+=SO3−1 from [5], and twisting methods. Once again, this is quite technical.
(3) A third method, leading to a quick and complete proof, is simply by using the ADE classification of the subgroups of S4+, from [5]. Indeed, a careful examination of the list in [5] shows that the subgroup < S4,Dd∞> in the statement can only be S4+ itself.
As an observation here, the method in the proof (3) of the above result applies to S5+ as well, by using the index 5 subfactor classification results from [19], [20].
We can now formulate our key technical result, as follows:
Theorem 6.5. The generation conjecture holds for quantum permutation groups.
Proof. This can be proved as follows:
(1) At N = 2,3 this is clear from SN =SN+.
(2) At N = 4 this is something that we know, from Proposition 6.4 above.
(3) At N ≥5 simplest is to use the following generation result, established in [14]:
SN+ =< SN, SN−1+ >
Indeed, this allows us to prove the generation property, by recurrence.
There are many interesting questions in relation with the above results. The main ques- tion is whether the recurrence argument from the proof of Theorem 6.5 can be replaced with something more direct, in the spirit of the formula in Proposition 6.3.
7. Liberation theory
We discuss here various liberation aspects, somehow in continuation to the stability results established in section 4 above, but at a more advanced level.
A first operation to be investigated is the half-liberation. In the orthogonal case, a standard result from [12], [13] states that any non-classical subgroupG⊂O∗N must appear via an antidiagonal symmetric 2×2 matrix model construction, denotedH →[H], from
a certain subgroup H ⊂ UN. As explained in [8], with this result in hand, G = [H], the corresponding standard tori are related by TQ(G) = [TQ(H)], and under suitable assumptions, this proves the various conjectures from section 2, in this case.
The situation in the general case G ⊂ UN∗ is quite similar. Indeed, we have a formula of type G = [[H]], with the operation H → [[H]] being given by an antidiagonal 2×2 matrix model, coming from [6], and then the arguments in [8] extend.
These results are of course useful if we want to investigate more in detail the various conjectures from section 2, and their generalizations. In relation with the generation conjecture, however, we can simply use here the recent approach in [3], and we have:
Proposition 7.1. Given a compact group TN ⊂GN ⊂ UN, where TN =ZN2 , the genera- tion conjecture holds for its half-liberation G∗N =< GN, TN∗ >.
Proof. This is trivial, because the half-liberation operation GN → G∗N, as formulated above, following [3], simply appears by enlarging the diagonal torus.
As a comment here, there is of course something non-trivial in all this, namely the fact that the construction GN →G∗N can be axiomatized as above. See [3].
Let us discuss now the liberation operation. According to [9], this operation associates to any easy group SN ⊂ GN ⊂ SN+ its free version SN+ ⊂ G+N ⊂ UN+, obtained at the Tannakian level by “removing the crossings”. This is perhaps a bit abstract, and at a more concrete level, it follows from the classification results in [24] that in the uniform case, where GN =GN ∩UN−1, the easy groups and their liberations are as follows:
ON, UN, BN, CN, HNs → O+N, UN+, B+N, CN+, HNs+
In relation with our generation question, however, the original formulation from [9], involving the “removal of the crossings”, is the one that we need. Indeed, in more modern terms, and more specifically in terms of the generation operation < , >, this amounts in setting G+N =< GN, SN+ >, as explained in [2]. With this picture in hand, we have:
Theorem 7.2. Given an easy compact group SN ⊂ GN ⊂UN, the generation conjecture holds for its free version G+N =< GN, SN+ >.
Proof. This is trivial from what we have already, namely from the stability property in Proposition 4.2, with Proposition 3.1 and Theorem 6.5 as ingredients.
Summarizing, with the modern approach from [2], [3] to the half-liberation and libera- tion operations in hand, everything becomes trivial, modulo Theorem 6.5.
8. Open problems
We have seen in this paper, and especially in the previous section, that the generation and intersection constructions ∩ and < , > allow a fresh approach to the maximal tori conjectures, and to the compact quantum groups in general.
As a first question, we have the unification of the methods used in Proposition 7.1 and Theorem 7.2, which are slightly different in nature, and then the application of these methods to the toral generation conjecture for more general intermediate liberations [22], [23]. As explained in [3], this amounts in developing “soft” and “hard” liberation methods for the compact Lie groups, as a generalization of [9] and subsequent work.
As a second question, we have the problem, already raised at the end of section 6, of understanding the generation property for SN+ via something more conceptual, uniform in N. This is in fact related to above-mentioned “soft” and “hard” liberation program for the compact Lie groups, which faces for the moment similar difficulties, namely the replacement of ad-hoc methods from [14], [15], [16] with something uniform in N.
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T.B.: Department of Mathematics, University of Cergy-Pontoise, F-95000 Cergy- Pontoise, France. [email protected]
