Liberation theory for noncommutative homogeneous spaces

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

TEODORBANICA

Liberation theory for noncommutative homogeneous spaces

Tome XXVI, n^o1 (2017), p. 127-156.

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Liberation theory for noncommutative homogeneous spaces

Teodor Banica⁽¹⁾

ABSTRACT. — We discuss the liberation question, in the homogeneous space setting. Our first series of results concerns the axiomatization and classification of the families of compact quantum groupsG= (GN) which are “uniform”, in a suitable sense. We study then the quotient spaces of typeX = (G_M ×G_N)/(G_L×G_M−L×GN−L), and the liberation operation for them, with a number of algebraic and probabilistic results.

RÉSUMÉ. — On étudie le problème de liberation, dans le cadre des espaces homogènes. Notre première série de résultats concerne l’axiomati- sation et la classification des familles de groupes quantiques compactsG= (GN) qui sont « uniformes », dans un sens convenable. On étudie ensuite les espaces quotient du typeX= (GM×GN)/(GL×GM−L×GN−L), et l’opération de liberation pour ces espaces, avec des résultats de nature algébrique et probabiliste.

Introduction

The notion of noncommutative space goes back to an old theorem of Gelfand, stating that any commutative C^∗-algebra must be of the form C(X), for a certain compact space X. In view of this result, one can de- fine the category of “noncommutative compact spaces” to be the category of C^∗-algebras, with the arrows reversed. The category of usual compact spaces embeds then covariantly into this category, viaX →C(X).

Once again by using the Gelfand theorem, each noncommutative space X can be thought of as appearing as “liberation” of its classical version

(*)Reçu le 14 décembre 2015, accepté le 7 mars 2016.

Keywords:Liberation theory, Homogeneous space.

Math. classification:46L65, 46L54.

(1) Département des Mathématiques, Cergy-Pontoise University, 95000 Cergy-Pontoise, France — [email protected]

Article proposé par Mireille Capitaine.

X_class, which is obtained by dividing the corresponding algebraC(X) by its commutator ideal.

We will be interested here in the liberation operation, in the algebraic manifold context. Given a family of noncommutative polynomials Pi ∈ Chz1, . . . , zNi, the associated noncommutative manifold X, and its classi- cal versionXclass, are given by:

X = Spec

C^∗

z₁, . . . , z_N

P_i(z₁, . . . , z_N) = 0

S S

Xclass = n

(z₁, . . . , zN)∈C^N

Pi(z₁, . . . , zN) = 0o

Here the family of polynomials {P_i} is assumed to be such that the biggestC^∗-norm on the universal∗-algebra hz₁, . . . , z_N|P_i(z₁, . . . , z_N) = 0i is bounded.

The liberation operationXclass→Xcan be axiomatized in the quantum group context, the idea being that the category of pairings, which encodes in an abstract way the commutation relations ab = ba, must be replaced by a new category of partitions. The theory here, based on Woronowicz’s fundamental work in [35], [36], on Wang’s free quantum groups [31], [32], on the Weingarten formula [4], [5], [12], [34], and on the liberation philosophy in free probability theory [9], [21], [27], [30], was developed in [8].

In the homogeneous space case, where the general study goes back to [10], [22], [23], some related theory, concerning spaces of typeX =GN/G_N−M, was developed in [6], and then in [7]. Such spaces were shown to have a number of interesting features, making them potential candidates for an algebraic manifold extension of [8], provided that the family of compact quantum groupsG= (GN) producing them satisfies:

(1) The “easiness” condition in [8], stating that we must haveSN ⊂GN, for anyN∈N, with these inclusions being of a certain special type.

(2) The “uniformity” condition, stating that we must have GN ∩U_N−M⁺ = GN−M, with respect to the standard embedding U_N−M⁺ ⊂U_N⁺.

We will review here this work, by using some new ideas, from [1], [2], [3]. On one hand, we will replace the easiness assumption by a condition of typeHN ⊂GN, which will allow us to use a twisting parameterq=±1. On the other hand, we will study more general quotient spaces, depending on

parametersL6M 6N, as follows:

X= (GM×GN)

(GL×GM−L×GN−L).

Our main result will be a verification of the Bercovici–Pata liberation criterion, for certain variables associatedχ∈C(X), in a suitableL, M, N →

∞limit.

There are many questions raised by the present work. Here are some of them:

(1) A first question concerns the full classification of the quantum groups satisfying the above conditions. For some recent advances here, see [15], [25], [28], [29].

(2) A second question concerns the validity in our setting of the quan- tum isometry group formula G⁺(X⁺) = G(X)⁺, in relation with the rigidity results in [11], [16].

(3) Yet another question regards the possible applications of the present formalism to free probability invariance questions, in the spirit of [13], [18].

(4) Finally, there are as well several interesting questions in relation with the axiomatization problem for the noncommutative algebraic manifolds [17].

Finally, regarding the general presentation of the paper:

(1) We use Woronowicz’s quantum group formalism in [35], [36], with the extra axiomS²=id. A reference here is the book [20]. Regarding the noncommutative homogeneous spaces, whose axiomatization is known to run into several difficulties [7], [10], [14], [22], [23], [26], we use here a simplified formalism, best adapted to our examples, that we intend to fully clarify in a forthcoming paper.

(2) The present work is a technical continuation of [1], [2], [3], [6], [7], with the aim of basically enlarging a list of examples. As already mentioned, the general theory is not available yet. We have therefore opted for an “example-first” presentation. This is actually in tune with the easy quantum group literature, where most of the basic examples were constructed and studied long before [8].

The paper is organized as follows: in Sections 1–2 we discuss the orthogonal and unitary cases, in Sections 3-4 we introduce all the quantum groups that we are interested in, and we study the associated homogeneous spaces, and in Sections 5–6 we discuss probabilistic aspects.

Acknowledgements

I would like to thank the referee for a careful reading of the manuscript, and for a number of useful suggestions. This work was partly supported by the “Harmonia” NCN grant 2012/06/M/ST1/00169.

1. Partial isometries

In this section and in the next one we discuss the construction of the ho- mogeneous spaces that we are interested in, in the case where the underlying groups or quantum groups areO_N, U_N, or their twists ¯O_N,U¯_N, or their free versionsO⁺_N, U_N⁺.

We begin with the classical case. Best is to start as follows:

Definition 1.1. — Associated to any integers L 6 M 6 N are the spaces

O_{M N}^L =n

T :E→F isometry

E⊂R^N, F ⊂R^M,dim_RE=Lo , U_{M N}^L =n

T :E→F isometry

E⊂C^N, F ⊂C^M,dim_CE=Lo , where the notion of isometry is with respect to the usual real/complex scalar products.

As a first observation, atL=M =Nwe obtain the groupsO_N, U_N. More generally, atM =N we obtain the various components of the semigroups Oe_N,Ue_N of partial isometries ofR^N,C^N, studied in [2], which are by definition given by:

Oe_N =

N

[

L=0

O_{N N}^L , Ue_N =

N

[

L=0

U_{N N}^L .

Yet another interesting specialization isL=M = 1. Here the elements ofO¹_1N are the isometriesT : E →R, with E ⊂R^N one-dimensional, and such an isometry is uniquely determined by the elementT⁻¹(1)∈R^N, which must belong to the sphereS^N−1

R . Thus, we haveO¹_1N =S^N−1

R . Similarly, in the complex case we haveU_1N¹ =S^N−1

C .

In general, the most convenient is to view the elements ofO_{M N}^L , U_{M N}^L as rectangular matrices, and to use matrix calculus for their study:

Proposition 1.2. — We have identifications of compact spaces O_{M N}^L 'n

U ∈MM×N(R)

U U^t=projection of traceLo , U_{M N}^L 'n

U ∈MM×N(C)

U U^∗=projection of traceLo ,

with each partial isometry being identified with the corresponding rectangular matrix.

Proof. — We can indeed identify the partial isometries T :E→F with their corresponding extensions U : R^N → R^M, U : C^N → C^M, obtained by setting U_E⊥ = 0, and then identify these latter linear mapsU with the

corresponding rectangular matrices.

As an illustration, at L = M = N we recover in this way the usual matrix description of O_N, U_N. More generally, at M = N we recover the usual matrix description of Oe_N,Ue_N. See [2]. Finally, at L = M = 1 we obtain the usual description ofS^N−1

R , S^N−1

C .

Now back to the general case, observe that the isometriesT:E→F, or rather their extensionsU :K^N →K^M, withK=R,C, obtained by setting U_E⊥ = 0, can be composed with the isometries ofK^M,K^N, according to the following scheme:

K^{N B}

∗ //K^{N U} //K^{M A} //K^M

B(E) //

OO

E ^T //

OO

F //

OO

A(F)

OO

In other words, the groups O_M ×O_N, U_M ×U_N act respectively on O^L_{M N}, U_{M N}^L . With the identifications in Proposition 1.2 made, the statement here is:

Proposition 1.3. — We have action maps as follows, which are tran- sitive,

O_M ×O_N yO^L_{M N} : (A, B)U =AU B^t, UM×UN yU_{M N}^L : (A, B)U =AU B^∗,

whose stabilizers are respectively O_L×O_M−L×O_N−L and U_L×U_M−L× U_N−L.

Proof. — We have indeed action maps as in the statement, which are transitive. Let us compute now the stabilizerGof the pointU = (¹₀ ⁰₀). Since the elements (A, B)∈GsatisfyAU =U B, their components must be of the

formA= (^x₀ ^∗_a), B= (^x_∗ ⁰_b). Now sinceA, Bare both unitaries, these matrices follow to be block-diagonal, and we obtain:

G=

(A, B) A=

x 0 0 a

, B= x 0

0 b

.

We conclude that the stabilizer ofU = (¹₀ ⁰₀) is parametrized by triples (x, a, b) belonging respectively toO_L×O_M−L×O_N−L andU_L×U_M−L×

U_N−L, as claimed.

Finally, let us work out the quotient space description ofO^L_{M N}, U_{M N}^L : Theorem 1.4. — We have isomorphisms of homogeneous spaces as fol- lows,

O_{M N}^L = (OM ×ON)/(OL×O_M−L×ON−L), U_{M N}^L = (U_M ×U_N)/(U_L×U_M−L×U_N−L),

with the quotient maps being given by(A, B)→AU B^∗, whereU = (¹₀ ⁰₀).

Proof. — This is just a reformulation of Proposition 1.3 above, by taking into account the fact that the fixed point used in the proof there was

U = (¹₀ ⁰₀).

Once again, the basic examples here come from the cases L= M =N andL=M = 1, where the quotient spaces at right are respectivelyON, UN

andON/ON−1, UN/UN−1. In fact, in the generalL=M case we obtain the following spaces, considered in [7]:

O_{M N}^M = (OM ×ON)/(OM ×ON−M) =ON/O_N−M, U_{M N}^M = (U_M ×U_N)/(U_M ×U_N−M) =U_N/U_N−M. For some further information on these spaces, we refer to [2], [7].

2. Liberations and twists

We discuss now some noncommutative versions of the above construc- tions. We use the quantum group formalism of Woronowicz [35], [36], with the extra axiomS²=id. In other words, we consider pairs (A, u) consisting of aC^∗-algebraA, and a unitary matrixu∈M_N(A), such that the following formulæ define morphisms ofC^∗-algebras:

∆(u_ij) =X

k

u_ik⊗u_kj, ε(u_ij) =δ_ij, S(u_ij) =u^∗_ji.

These morphisms are called comultiplication, counit and antipode. We write A = C(G), and call G a compact matrix quantum group. For full details here, see [20].

We recall from [1], [5] that the compact groupsON, UN can be twisted, by replacing the commutation relationsab=ba, ab^∗=b^∗abetween the standard coordinates uij(g) =gij with the following commutation/anticommutation relations:

ab^×=

(−b^×a fora6=b on the same row or column ofu, b^×a otherwise.

Hereb^× =b, b^∗, and the precise statement is that these relations, when applied to a matrixu= (uij) which is orthogonal (u= ¯u, u^t=u⁻¹, where

¯

u= (u^∗_ij)), respectively biunitary (u^∗ = u⁻¹, u^t = ¯u⁻¹) produce quantum groups ¯O_N,U¯_N. See [1], [5].

We can liberateO^L_{M N}, U_{M N}^L , and then twist them, as follows:

Definition 2.1. — Associated to any integers L 6 M 6 N are the algebras

C(O_{M N}^L+ ) =C^∗

(u_ij)i=1,...,M,j=1,...,N

u= ¯u, uu^t=projection of trace L , C(U_{M N}^L+) =C^∗

(uij)i=1,...,M,j=1,...,N

uu^∗,uu¯ ^t=projections of trace L , and their quotients C( ¯O^L_{M N}), C( ¯U_{M N}^L ), obtained by imposing the twisting relations.

Observe that the above universal algebras are well-defined, because the trace conditions, which read P

ijuiju^∗_ij = P

iju^∗_ijuij = L, show that we have||uij||6√

L.

We have inclusions between the various spaces constructed so far, as follows:

U_{M N}^L //U_{M N}^L+ oo U¯_{M N}^L

O^L_{M N} //

OO

O^L+_{M N}

OO

O¯^L_{M N}

oo OO

Indeed, the inclusions at right follow from definitions, and those at left come from Proposition 1.2, and from the fact thatO^L_{M N}, U_{M N}^L are stable by conjugation.

At the level of basic examples now, we first have the following result:

Proposition 2.2. — AtL=M = 1we obtain the diagram S^N−1

C //S^N−1

C,+ S¯^N−1

oo C

S^N−1

R //

OO

S^N−1

R,+

OO

S¯^N−1

oo ROO

consisting of the liberations and twists of the spheresS^N−1

R , S^N−1

C .

Proof. — We recall from [1] that the various spheres are constructed as follows, with the symbol×standing for “commutative”, “twisted” and “free”, respectively:

C(S^N−1

R,× ) =C_×^∗

z₁, . . . , z_N

z_i=z^∗_i,X

i

z²_i = 1

, C(S^N−1

C,× ) =C_×^∗

z1, . . . , zN

X

i

ziz_i^∗=X

i

z^∗_izi= 1

.

Now by comparing with the definition ofO^1×_1N, U_1N^1×, this proves our claim.

We have as well the following result, once again making the link with [1]:

Proposition 2.3. — At L=M =N we obtain the diagram U_N //U_N⁺ oo U¯_N

ON //

OO

O_N⁺

OO

O¯N

oo OO

consisting of the liberations and twists of the groupsON, UN.

Proof. — We recall from [1] that the various quantum groups are con- structed as follows, with the symbol×standing once again for “commuta- tive”, “twisted” and “free”:

C(O^×_N) =C_×^∗

(u_ij)i,j=1,...,N

u= ¯u, uu^t=u^tu= 1 , C(U_N^×) =C_×^∗

(uij)i,j=1,...,N

uu^∗=u^∗u= 1,uu¯ ^t=u^tu¯= 1 .

On the other hand, according to Proposition 1.2 and to Definition 2.1 above, we have the following presentation results:

C(O^N_{N N}^×) =C_×^∗

(uij)i,j=1,...,N

u= ¯u, uu^t= projection of traceN , C(U_{N N}^N×) =C_×^∗

(uij)i,j=1,...,N

uu^∗,uu¯ ^t= projections of traceN . We use now the standard fact that ifp=aa^∗ is a projection thenq = a^∗a is a projection too. Together with T r(uu^∗) = T r(u^tu) and¯ T r(¯uu^t) = T r(u^∗u), this gives:

C(O^N_{N N}^×) =C_×^∗

(u_ij)i,j=1,...,N

u= ¯u, uu^t, u^tu= projections of traceN , C(U_{N N}^N×) =C_×^∗

(u_ij)i,j=1,...,N

uu^∗, u^∗u,uu¯ ^t, u^tu¯= projections of traceN . Now observe that, in tensor product notation, and by using the normal- ized trace, the conditions at right are all of the form (tr⊗id)p = 1, with p=uu^∗, u^∗u,uu¯ ^t, u^tu. We therefore obtain (tr¯ ⊗ϕ)(1−p) = 0 for any faith- ful state ϕ, and it follows that the projectionsp=uu^∗, u^∗u,uu¯ ^t, u^tu¯ must

be all equal to the identity, as desired.

Regarding now the homogeneous space structure ofO^L×_{M N}, U_{M N}^L×, the sit- uation here is more complicated in the twisted and free cases than in the classical case. See [1], [7].

The classical results have, however, some partial extensions. In order to formulate a result, we use the standard coaction formalism for the compact quantum groups, as in [7]. Also, given two noncommutative compact spaces X, Y, we define their productX×Y via the formula C(X×Y) =C(X)⊗ C(Y), with the tensor product being the maximal one. Finally, we use the standard fact that whenG, Hare compact matrix quantum groups, then so is their productG×H. For more on these topics, see [7], [20].

With these conventions, we have the following result:

Proposition 2.4. — The spacesU_{M N}^L× have the following properties:

(1) We have an action U_M^× ×U_N^× yU_{M N}^L×, given by uij →P

klaik⊗ b^∗_jl⊗u_kl.M

(2) We have a mapU_M^× ×U_N^× →U_{M N}^L×, given byuij→P

l6Lail⊗b^∗_jl. Similar results hold for the spacesO^L×_{M N}, with all the∗ exponents removed.

Proof. — In the classical case, the transpose of the action map U_M × U_N y U_{M N}^L and of the quotient map U_M ×U_N → U_{M N}^L are as follows, whereJ = (¹₀ ⁰₀):

ϕ → ((A, B, U)→ϕ(AU B^∗)), ϕ → ((A, B)→ϕ(AJ B^∗)).

But withϕ=u_ij we obtain precisely the formulæ in the statement. The proof in the orthogonal case is similar. Regarding now the free case, the proof goes as follows:

(1) Assuminguu^∗u=u, with Uij=P

klaik⊗b^∗_jl⊗ukl we have:

(U U^∗U)ij=X

pq

X

klmnst

aika^∗_qmaqs⊗b^∗_plbpnb^∗_jt⊗uklu^∗_mnust,

= X

klmt

a_ik⊗b^∗_jt⊗u_klu^∗_mlu_mt=X

kt

a_ik⊗b^∗_jt⊗u_kt=U_ij.

Also, assuming that we haveP

iju_iju^∗_ij=L, we obtain:

X

ij

UijU_ij^∗ =X

ij

X

klst

aika^∗_is⊗b^∗_jlbjt⊗uklu^∗_st=X

kl

1⊗1⊗uklu^∗_kl =L .

(2) Assuminguu^∗u=u, with Vij =P

l6Lail⊗b^∗_jl we have:

(V V^∗V)_ij =X

pq

X

x,y,z6L

a_ixa^∗_qya_qz⊗b^∗_pxb_pyb^∗_jz =X

x6L

a_ix⊗b^∗_jx=V_ij.

Also, assuming that we haveP

iju_iju^∗_ij=L, we obtain:

X

ij

VijV_ij^∗=X

ij

X

l,s6L

aila^∗_is⊗b^∗_jlbjs=X

l6L

1 =L .

By removing all the∗exponents, we obtain as well the orthogonal results.

In the twisted case the proof is similar. Let us first discuss the orthogonal case. The twisting relations can be written as follows:

u_iju_pq= (−1)^δip+δjqu_pqu_ij.

With this formula in hand, the verification of the extra relations goes as follows:

(1) WithU_ij =P

kla_ik⊗b_jl⊗u_kl we have, as desired:

U_ijU_pq= X

klmn

a_ika_pm⊗b_jlb_qn⊗u_klu_mn

= X

klmn

(−1)^δip+δkma_pma_ik⊗(−1)^δjq+δlnb_qnb_jl⊗(−1)^δkm+δlnu_mnu_kl

= X

klmn

(−1)^δip+δjqa_pma_ik⊗b_qnb_jl⊗u_mnu_kl= (−1)^δip+δjqU_pqU_ij.

(2) WithVij =P

l6Lail⊗bjl we have as well, as desired:

VijVpq= X

l,m6L

ailapm⊗bjlbqm

= X

l,m6L

(−1)^δip+δlmapmail⊗(−1)^δjq+δlmbqmbjl

= X

l,m6L

(−1)^δip+δjqapmail⊗bqmbjl= (−1)^δip+δjqVpqVij.

The proof in the unitary case is similar, by adding ∗ exponents where

needed.

Let us examine now the relation between the above maps. In the classical case, given a quotient spaceX =G/H, the associated action and quotient maps are given by:

(a:G×X →X : (g, g⁰H)→gg⁰H p:G→X : g→gH

Thus we havea(g, p(g⁰)) =p(gg⁰). In our context, a similar result holds:

Theorem 2.5. — With G =G_M ×G_N and X = G^L_{M N}, where G_N = O^×_N, U_N^×, we have

G×G ^m //

id×p

G

p

G×X ^a //X

wherea, pare the action map and the map constructed in Proposition 2.4.

Proof. — At the level of the associated algebras of functions, we must prove that the following diagram commutes, where Φ, π are morphisms of

algebras induced bya, p:

C(X) ^Φ //

π

C(G×X)

id⊗π

C(G) ^∆ //C(G×G)

When going right, and then down, the composition is as follows:

(id⊗π)Φ(uij) = (id⊗π)X

kl

aik⊗b^∗_jl⊗ukl =X

kl

X

s6L

aik⊗b^∗_jl⊗aks⊗b^∗_ls.

On the other hand, when going down, and then right, the composition is as follows, whereF23is the flip between the second and the third components:

∆π(uij) =F23(∆⊗∆)X

s6L

ais⊗b^∗_js=F23

X

s6L

X

kl

aik⊗aks⊗b^∗_jl⊗b^∗_ls

! .

Thus the above diagram commutes indeed, and this gives the result.

In general, going beyond Theorem 2.5 leads to some non-trivial questions.

A first issue comes from the fact that the inclusionsG_L×G_M−L×G_N−L⊂ G_M ×G_N are not well-defined, in the free case. There are as well some analytic issues, coming from the fact that the maps in Proposition 2.4(2) are in general not surjective. See [1], [7].

3. Uniform quantum groups

We discuss in this section a generalization of the above constructions. For this purpose, we first need to axiomatize a suitable class of compact quantum groups, generalizing the classical groupsO_N, U_N, their twists ¯O_N,U¯_N, and their free versionsO_N⁺, U_N⁺.

LetP(k, l) the set of partitions between an upper row of k points, and a lower row of l points, with each leg colored black or white, and withk, l standing for the corresponding “colored integers”. We have the following no- tion, which appears as a straightforward generalization of the corresponding orthogonal notion from [8]:

Definition 3.1. — A category of partitions is a collection of setsD= S

klD(k, l), withD(k, l)⊂P(k, l), which contains the identity, and is stable under:

(1) The horizontal concatenation operation⊗.

(2) The vertical concatenation◦, after deleting closed strings in the mid- dle.

(3) The upside-down turning operation∗ (with reversing of the colors).

Here the vertical concatenation operation assumes of course that the colors match. Regarding the identity, the precise condition is that D(◦,◦) contains the “white” identity^◦_◦^|. By using (3) we see thatD(•,•) contains the

“black” identity^•_•^|, and then by using (1) we see that eachD(k, k) contains its corresponding (colored) identity.

The basic example of such a category isP itself. Yet another basic exam- ple is the categoryN C of noncrossing partitions. There are of course many other examples. We refer to [8] for the uncolored case, and to [28], [29] for the general colored case.

As explained in [8], [19], [29], such categories produce quantum groups.

In this paper, however, we will need a modification of this construction, in order to cover as well the twists. We use for this purpose a number of findings from our recent papers [1], [3].

As explained in [1], [3], in order to cover the twists of the quantum groups in [8] we must restrict attention to the categoriesD⊂Peven, wherePeven⊂ P is the category of partitions having all blocks of even size. Such partitions act on tensors, as follows:

Definition 3.2. — Associated to anyπ∈P_even(k, l)are the linear maps T_π(e_i1⊗. . .⊗e_ik) = X

j:ker(ⁱ_j)6π

e_j1⊗. . .⊗e_jl, T¯π(ei₁⊗. . .⊗ei_k) =X

τ6π

ε(τ) X

j:ker(ⁱ_j)=τ

ej₁⊗. . .⊗ej_l,

where {ei} is the standard basis of C^N, and ε : P_even → {−1,1} is the signature map.

Here the kernel of a multi-index (ⁱ_j) = (ⁱ_j^1...ik

1...j_l) is the partition obtained by joining the sets of equal indices. Thus, the condition ker(ⁱ_j)6π simply tells us that the strings ofπ must join equal indices. As for the signature mapε:Peven→ {−1,1}, this is a canonical extension of the usual signature mapε:S_∞→ {−1,1}, constructed in [1].

Here are a few examples of such linear maps, taken from [1]:

T_∩(1) = ¯T_∩(1) =X

i

ei⊗ei, T_∪(ei⊗ej) = ¯T_∪(ei⊗ej) =δij ,

T_/\(e_i⊗e_j) =e_j⊗e_i, T¯_/\(e_i⊗e_j) =

(ej⊗ei fori=j ,

−ej⊗ei fori6=j .

In general,Tπ,T¯πcan be thought of as coming from a twisting parameter q=±1. As explained in [1], for π∈N Ceven we haveτ 6π =⇒ ε(τ) = 1, and soT_π= ¯T_π. In general, however, the mapsT_π,T¯_πare different. We refer to [1] for full details here.

We have the following “q-easiness” notion, inspired from [8]:

Definition 3.3. — A compact quantum group G⊂U_N⁺ is called quizzy when

Hom(u^⊗k, u^⊗l) =span T˙π

π∈D(k, l) ,

for any colored integersk, l, for a certain category of partitionsD⊂P_even. Here the dot stands for a fixed value ofq=±1, with the maps T_π being used atq= 1, and with the maps ¯T_π being used atq=−1. Also, the “col- ored” tensor powersu^⊗k, u^⊗lare defined by tensoring the corepresentations u^◦=uandu^•= ¯u.

At the level of basic examples, we have the following result:

Proposition 3.4. — The following are quizzy quantum groups, UN //U_N⁺ oo U¯N

O_N //

OO

O_N⁺

OO

O¯_N

oo OO

with q ∈ {−1,1} being by definition 1 at left, −1 at right, and ±1 in the middle.

Proof. — As explained in [1], [8], the above quantum groups appear in- deed from the following categories of partitions, with q∈ {−1,1} being as

in the statement:

P2

N C2

//

oo P2

P₂ N C₂oo //P₂

Here P₂ is the set of all pairings, P₂ ⊂ P₂ is the set of “matching”

pairings, whose upper and lower strings connect ◦−•, and whose through strings connect◦−◦or •−•, andN C2,N C2 are the corresponding subsets of

noncrossing pairings. See [1], [8].

Consider now the groupH_N^s =ZsoSN, withs∈ {2,4, . . . ,∞}, which con- sists of the permutation matricesσ∈SN with nonzero entries multiplied by elements ofZs. This group has a free analogue,H_N^s+=Zso∗S_N⁺, constructed in [4], as follows:

C(H_N^s+) =C^∗

(u_ij)i,j=1,...,N

u_iju^∗_ij =u^∗_iju_ij=p_ij= magic, u^s_ij =p_ij .

Here the “magic” condition states that the entries ofp= (p_ij) are pro- jections, summing up to 1 on each row and column, and the last condition, u^s_ij = pij, disappears by definition at s = ∞. Observe that the classical version ofH_N^s+ is indeedH_N^s. See [4].

Thes= 2,∞specializations of H_N^s, H_N^s+, denoted respectively HN, H_N⁺ andK_N, K_N⁺, are the quantum groups in [5], and their complex analogues.

We have:

Proposition 3.5. — The following are quizzy quantum groups, at both q=±1,

KN //K_N⁺

H_N^s //

OO

H_N^s+

OO

H_N //

OO

H_N⁺

OO

whereH_N^s =ZsoSN, and whereH_N^s+=Zso_∗S_N⁺, withs∈ {2,4, . . . ,∞}.

Proof. — As explained in [3], [4], the above quantum groups appear in- deed, with parameter q=±1 as in the statement, from the following cate- gories of partitions:

Peven

N Ceven

oo

P_even^s

N C_even^s

oo

Peven oo N Ceven

HereP_even^s ⊂P_even is the set of partitions having the property that, in each block, the number of white legs equals the number of black legs, modulo s, with all legs counted with coefficient + up, and−down. At right we have the subsetN C_even^s =P_even^s ∩N C, and the lower and upper objects are the

corresponding specializations ats= 2,∞.

We recall that the free complexification (G,e u) of a compact matrix quan-e tum group (G, u) is obtained by considering the subalgebra C(G)e ⊂C(T)∗ C(G) generated by the entries ofeu=zu, wherezis the standard generator ofC(T). See [24].

With this convention, we have the following extra example, from [29]:

Proposition 3.6. — The quantum group K_N⁺⁺ = Ke_N⁺ appears as an intermediate object

K_N⁺ ⊂K_N⁺⁺ ⊂U_N⁺,

with both inclusions being proper, and is quizzy, equal to its own twist.

Proof. — By composing the canonical inclusionC(K_N⁺⁺)⊂C(T)∗C(K_N⁺) withε∗id we obtain a morphismC(K_N⁺⁺)→C(K_N⁺) mappingue_ij → u_ij, so we have inclusions as in the statement. Since the elementsp_ij=ue_ijue^∗_ij= zu_iju^∗_ijz^∗ andq_ij=ue^∗_ijue_ij =u^∗_iju_ij are projections, and are not equal, both these inclusions are proper.

Regarding the easiness claim, this follows from the general theory of the representations of free complexifications [24]. To be more precise, as ex- plained in [29], the associated categoryN C⁻_evenis that of the even noncross- ing partitions which, when rotated on one line, have alternating colors in each block. Observe that the inclusions in the statement correspond then to the inclusions at the partition level, which are as follows:

N Ceven⊃ N C⁻_even⊃ N C2,

Finally, sinceN C⁻_even⊂N C_even, the quantum groupK_N⁺⁺equals its own

twist.

The above examples are in fact the only ones that we are interested in, in the classical/twisted and free cases. In order to axiomatize these objects, we use:

Proposition 3.7. — For a quizzy quantum group HN ⊂ GN ⊂ U_N⁺, coming from a category of partitions N C2 ⊂D ⊂Peven, the following are equivalent:

(1) D is stable by removing blocks.

(2) GN ∩U_N−M⁺ =GN−M, for anyM 6N.

If these conditions are satisfied, we call bothG= (GN)andD “uniform”.

Proof. — This was proved in [7] in the orthogonal case, and the proof in general is similar. Assume that we have a subgroupK ⊂U_N−M⁺ , with fun- damental representationv, and consider theN×N matrix ˜v=diag(v,1M).

Then, for anyπ∈Peven, we have:

Tπ∈Hom(˜v^⊗k,v˜^⊗l) ⇐⇒ Tπ⁰ ∈Hom(v^⊗k0, v^⊗l0),∀π⁰⊂π .

With this formula in hand, we deduce that given a subgroupG ⊂U_N⁺, with fundamental representation denotedu, the algebra of functions onK= G∩U_N⁺_−M is given by:

C(K) =C(U_N⁺_−M).

T ∈Hom(˜v^⊗k,v˜^⊗l),∀T ∈Hom(u^⊗k, u^⊗l) .

Thus, we haveGN∩U_N⁺_−M =G⁰_N−M, whereG⁰ = (G⁰_N) is the easy quan- tum group associated to the categoryD⁰ generated by all the subpartitions of the partitions inD. In particularGN ∩U_N⁺_−M =GN−M for anyM 6N

is equivalent toD=D⁰, as claimed.

Observe that the quantum groups in Propositions 3.4, 3.5 and 3.6 are all uniform. We have in fact the following result, where by “classical/twisted”

and “free” we mean /\ ∈ D and D ⊂ N Ceven, where D ⊂ Peven is the associated category of partitions:

Theorem 3.8. — The uniform classical / twisted and free quantum groups are

U_N,U¯_N

))K_N //00

K_N⁺ //K_N⁺⁺ //U_N⁺

H_N^s //

OO

H_N^s+

OO

HN //

OO ..

H_N⁺ //

OO

O⁺_N

OO

ON,O¯N

55

whereH_N^s =ZsoS_N,H_N^s+ =Zso_∗S_N⁺, with s∈ {2,4, . . . ,∞}, and K_N⁺⁺= Ke_N⁺.

Proof. — This is a consequence of the recent classification results in [28], [29], the idea being as follows. First, the diagram in the statement being obtained by merging the examples in Propositions 3.4, 3.5 and 3.6, all the above quantum groups are quizzy.

The uniformity condition is clear as well, for each of the quantum groups under consideration. Finally, all these quantum groups are either classi- cal/twisted or free.

In order to prove now the converse, in view of the twisting results in [2], it is enough to deal with the q = 1 case. So, consider a uniform category of partitions D ⊂Peven, as in Proposition 3.7. We must prove that in the classical/free cases, the solutions are:

P2

uuPeven

N Ceven

oo

N C⁻_even

oo N C2oo

oo

P_even^s

N C_even^s

oo

Pevenoo N Ceven N C2

oo oo

P2

ii

To be more precise, in the classical case, where/\∈D, we must prove that the only solutions are the categoriesP2,P2, P_even^s , and that in the free case, whereD⊂N Ceven, we must prove that the only solutions are the categories N C2,N C2,N C⁻_even, N C_even^s .