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A NOTE ON THE HALF-LIBERATION OPERATION
Teo Banica
To cite this version:
Teo Banica. A NOTE ON THE HALF-LIBERATION OPERATION. 2019. �hal-02098545�
TEO BANICA
Abstract. We propose a new approach to the half-liberation question, for the compact groups TN ⊂GN ⊂UN, whereTN =ZN2. Indeed, we can construct a quantum group TN∗ ⊂ G∗N ⊂ UN∗, simply by setting G∗N =< GN, TN∗ >. We explain here how this construction fits into the known general theory of half-liberation, and we discuss as well some potential generalizations, withTN∗ being replaced by more complicated objects.
CONTENTS
Introduction 1
1. Soft exit 2
2. Hard exit 3
3. General theory 4
4. Open problems 5
References 6
Introduction
An algebraic quantum group or algebraic noncommutative manifold is called half- classical when its coordinates are subject to the half-commutation relations abc = cba.
These relations relax the usual commutation relations ab=ba. Quite remarkably, under very strong assumptions, these relations are “unique”, in the sense that the half-classical world is the only one, between the classical one and the free one. See [5], [8].
The half-commutation relations automatically hold for the 2×2 antidiagonal matrices, and conversely, the various half-classical objects can be generally modelled by using 2×2 antidiagonal matrices. A lot of theory can be developed, based on this general principle.
To be more precise, the case of the half-classical orthogonal quantum groupsG⊂ON∗ was discussed in [11], the case of the compact quantum subspaces X ⊂ SN−1
R,∗ was discussed in [10], and some complex extensions of these results were discussed in [3].
Summarizing, the general half-classical theory is somewhat complete, with [3], [5], [8], [10], [11] and of course [18] providing a solid basis, for any further investigation.
2010Mathematics Subject Classification. 46L65 (20G42).
Key words and phrases. Compact quantum group, Half-liberation.
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2 TEO BANICA
However, the half-liberation operation itself, GN → G∗N for the quantum groups and XN →XN∗ for the manifolds, remains a bit mysterious. We propose here an answer to this question, in the case of the compact groups TN ⊂GN ⊂UN, whereTN =ZN2 . Indeed, we can construct a compact quantum group TN∗ ⊂G∗N ⊂UN∗, simply by setting:
G∗N =< GN, TN∗ >
Our goal here will be that of showing that this simple construction appears as a good complement to the general quantum group theory from [3], [11]. This is of course some- thing quite specialized, strictly dealing with the quantum group case. We will comment as well on the possibility of replacingTN∗ with more complicated liberations of TN.
The paper is organized as follows: in 1-2 we present our half-liberation operation, first in a “soft” form, using HN∗, and then in the “hard” form, usingTN∗, and in 3-4 we discuss the general theory, and some potential generalizations and open problems.
1. Soft exit
Let HN =Z2 oSN be the hyperoctahedral group, and consider as well its free version HN+ = Z2o∗SN+. By imposing the relations abc =cba to the standard coordinates of HN+ we obtain an intermediate quantum group HN ⊂HN∗ ⊂HN+. See [11].
We can exit the world of classical groups very quickly, as follows:
Definition 1.1. The half-liberation of an intermediate compact group HN ⊂ GN ⊂ UN is the intermediate compact quantum group HN ⊂GN ⊂UN given by
G∗N =< GN, HN∗ >
with the generation operation being taken in a topological sense, as an operation for the closed subgroups of the free unitary quantum group UN+.
This definition is something new, although quite folklore, and very simple. We should mention however that the generation operation, which goes back to [14], is not exactly something simple. For a review of the known properties of < , >, we refer to [4].
As a main result regarding this operation, we have:
Theorem 1.2. The half-liberations of the uniform easy groups, namely ON, UN , HN2 =HN, HN4, HN6, . . . , HN∞=KN
coincide with their usual half-liberations, taken in the easy sense, namely O∗N, UN∗ , HN2∗ =HN∗, HN4∗, HN6∗, . . . , HN∞∗=KN∗ obtained by liberating, and then by imposing the relations abc=cba.
Proof. This is something quite well-known. First of all, it follows from [17] that the easy compact groupsSN ⊂GN ⊂UN satisfying the uniformity assumptionGN−1 =GN∩UN−1+ are precisely those in the statement, with HNs =ZsoSN.
In order to compute the half-liberations in our sense, we use the fact that the operations
< , > and ∩ are “dual” to each other via Tannakian duality G↔C, as follows:
C<G,H>=CG∩CH , CG∩H =< CG, CH >
With standard easy quantum group notations, if we denote by D the category of par- titions for GN, and by G×N the easy half-liberation of GN, we have then:
CG∗
N = CGN ∩CH∗
N
= span(D)∩span(Peven∗ )
= span(D∩Peven∗ )
= span(D∩N Ceven, /|\)
= CG×
N
Here all the equalities are well-known and standard. Thus G∗N =G×N, as claimed.
Summarizing, we have so far a notion of half-liberation for the intermediate compact groups HN ⊂GN ⊂UN, which works well in the easy case.
2. Hard exit
In this section we discuss a modified version of our half-liberation operationGN →G∗N, which is more general, a bit harder to compute, and which will be our standard one.
Consider the group TN = ZN2 , with the notation standing for the fact that this group can be identified with the standard cube of RN, and is therefore a “real torus”.
Consider as well the half-classical version TN∗ = Zd◦N2 of this real torus, where ◦ is the half-classical product of discrete groups, subject to the relations abc = cba between the standard generators. The notation comes from the fact that we cannot use here the symbol ∗, which is reserved for the free product of discrete groups.
We can extend Definition 1.1 above, as follows:
Definition 2.1. The half-liberation of an intermediate compact groupTN ⊂GN ⊂UN is the intermediate compact quantum group TN∗ ⊂GN ⊂UN given by
G∗N =< GN, TN∗ >
with the generation operation being taken as usual in a topological sense.
Our first task is to verify that this more general notion is compatible with the one that we already have, introduced and studied in section 1 above.
This is something non-trivial, and we have indeed:
Theorem 2.2. For an intermediate compact group HN ⊂ GN ⊂ UN, its “soft” and
“hard” half-liberations, from Definition 1.1 and Definition 2.1, coincide.
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Proof. We must prove that for any intermediate compact groupHN ⊂GN ⊂UN, we have the following equality, between closed subgroups of UN+:
< GN, HN∗ >=< GN, TN∗ >
It is enough to solve the problem for the smallest possible group under consideration, namely GN =HN. Thus, we are led into the following question:
HN∗ =< HN, TN∗ >
Let G ↔ C be the Tannakian correspondence, with C = (Hom(u⊗k, u⊗l)), let G → P G be the projective version construction, and let (G ⊂ UN) → ([G] ⊂ ON∗) be the construction in [11]. We have then, by using a number of standard facts:
P HN∗ =P < HN, TN∗ > ⇐⇒ P KN =P < HN, TN∗ >
⇐⇒ CP KN =CP <HN,T∗
N>
⇐⇒ CP KN =CP HN ∩CP T∗
N
⇐⇒ CP KN =CP HN ∩CPTN
⇐⇒ CP KN =CP <HN,TN>
⇐⇒ CP KN =CP KN
Thus the projective versions coincide, and so the affine lifts must coincide as well.
As a comment here, the above proof is not the only one. Since HN, HN∗ are both easy, coming from Peven, Peven∗ , our question HN∗ =< HN, TN∗ >becomes:
span(Peven∗ ) =span(Peven)∩CT∗
N
But this can be proved by standard combinatorics, based on the fact, from [5], [8], that the half-classical combinatorics comes from the infinite symmetric groupS∞.
Summarizing, we have now a notion of half-liberation for the intermediate compact groups TN ⊂GN ⊂UN, which works well in the easy case.
3. General theory
We discuss now the compatibility with the general theory in [3], [11]. There are many potential things to be done here, and we will focus on the essential ones.
We first discuss the orthogonal case. We recall from [11] that the closed subgroups of ON∗ appear from the closed subgroups of UN via a matrix model operation EN → [EN], involving self-adjoint antidiagonal matrices. Now since any half-liberationG∗N in our sense is indeed a closed subgroup of ON∗ , we should therefore haveG∗N = [EN], for some closed subgroup EN ⊂UN, which remains to be explicitely computed.
The answer to this question is very simple, as follows:
Proposition 3.1. For any compact group TN ⊂GN ⊂UN we have the formula G∗N = [G◦N]
where G◦N =< GN,TN >, and where EN →[EN] is the construction in [11].
Proof. This can be proved by using the same method as for Theorem 2.2 above. With the notations from there, we have the following computation:
P G∗N =P[G◦N] ⇐⇒ P G∗N =P G◦N
⇐⇒ P < G∗N, TN∗ >=P < GN,TN >
⇐⇒ CP <G∗
N,TN∗>=CP <GN,TN>
⇐⇒ CP GN ∩CP T∗
N =CP GN ∩CPT∗
N
⇐⇒ CP GN ∩CP T∗
N =CP GN ∩CP T∗
N
Thus the projective versions coincide, and so the affine lifts must coincide as well.
In the unitary case, the situation is similar. We recall from [3] that the closed subgroups ofUN∗ appear from the closed subgroups ofUN via a matrix model operationEN →[[EN]], involving antidiagonal matrices. Now since any half-liberation G∗N in our sense is indeed a closed subgroup ofUN∗, we should therefore have G∗N = [[EN]], for some closed subgroup EN ⊂UN, which remains to be explicitely computed.
The answer to this question is once again very simple, as follows:
Theorem 3.2. For any compact group TN ⊂GN ⊂UN we have the formula G∗N = [[G◦N]]
where G◦N =< GN,TN >, and where EN →[[EN]] is the construction in [3].
Proof. The computation here is identical with the one in the proof of Proposition 3.1,
with technical ingredients coming this time from [3].
Summarizing, our notion of half-liberation fits perfectly with [3], [11]. It is of course possible to develop some more general theory, based on [3], [11] and related papers, but we will stop here, having basically said what we had to say.
4. Open problems
A first related question concerns the soft and hard liberation of the compact Lie groups, generalizing [7]. The soft liberation is related to the notion of easy envelope, from [1].
The hard liberation, however, requires combinatorial computations in the spirit of the one discussed at the end of section 2, or recurrence technology from [12], [13], [14]. There is in addition a problem with the diagonal torus of SN+, which collapses, but this can be solved by using standard maximal torus theory from [6], [9].
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A second related question concerns using as “input” more complicated liberations of HN, or of TN. The first thought here goes to the intermediate easy quantum groups HN ⊂HN× ⊂HN+, fully classified in [16]. As an example of what can be done here, the soft intermediate liberation ofKN =ToSN leads to quantum groupsKN ⊂KNΓ ⊂KN[r]⊂KN+, which together with [8], [15], [16] reasonably fill the vertical edges of the cube in [2].
A third related question regards the noncommutative spheres, and other algebraic mani- folds, studied in [10], and then in [3], by using the original methods from [11]. The problem is whether our present methods can be adapted as to cover such manifolds. This looks quite difficult, the operation< , > being very quantum group-specific.
References
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Soc.95 (2017), 519–540.
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[7] T. Banica and R. Speicher, Liberation of orthogonal Lie groups,Adv. Math.222(2009), 1461–1501.
[8] T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60(2010), 2137–2164.
[9] J. Bichon, Algebraic quantum permutation groups, Asian-Eur. J. Math.1(2008), 1–13.
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[12] M. Brannan, A. Chirvasitu and A. Freslon, Topological generation and matrix models for quantum reflection groups, preprint 2018.
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T.B.: Department of Mathematics, University of Cergy-Pontoise, F-95000 Cergy- Pontoise, France. [email protected]
