Topological generation and matrix models for quantum reflection groups
aa r X i v : . [ m a t h . OA ] A ug Topological generation and matrix models for quantum reflectiongroups
Michael Brannan, Alexandru Chirvasitu, Amaury Freslon
Abstract
We establish several new topological generation results for the quantum permutation groups S + N and the quantum reflection groups H s + N . We use these results to show that these quantumgroups admit sufficiently many “matrix models”. In particular, all of these quantum groups haveresidually finite discrete duals (and are, in particular, hyperlinear), and certain “flat” matrixmodels for S + N are inner faithful. Key words: quantum permutation group, quantum reflection group, compact quantum group, dis-crete quantum group, residually finite, hyperlinear, matrix model
MSC 2010: 20G42; 46L52; 16T20
Contents ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 References 26
The central objects of study in this paper are quantum permutations and their generalizations, quantum reflections . Given N ∈ N and an Hilbert space H , an N × N matrix P = [ P ij ] ≤ i,j ≤ N ∈ N ( B ( H )) ( B ( H ) being the C ∗ -algebra of bounded linear operators on H ) is called a quantumpermutation matrix (or magic unitary ) if its entries P ij are self-adjoint projections satisfying therelations P i P ij = 1 B ( H ) = P j P ij for each 1 ≤ i, j ≤ N .The simplest examples of quantum permutation matrices are of course the classical permuta-tion matrices (which correspond to those quantum permutations P associated to a one-dimensionalHilbert space H ). In fact, more generally any quantum permutation P with commuting entries { P ij } i,j corresponds to a subset X ⊆ S N of permutation matrices. Indeed, in this case the com-mutative C ∗ -algebra C ∗ ( { P ij } i,j ) generated by the P ij ’s is by Gelfand duality isomorphic to C ( X ),the C ∗ -algebra of complex functions on some subset X ⊆ S N . In particular P is identified this waywith the identity function on X ⊂ S N ⊂ M N ( C ). On the other hand, if one now considers quantumpermutations P whose entries do not commute, the structure of these objects becomes much lesswell-understood. From an operator algebraic point of view, this should come as no surprise, as theC ∗ -algebras C ∗ ( P ij | ≤ i, j ≤ N ) ⊂ B ( H ) generated by the entries of a quantum permutationmatrix P with non-commuting entries can be highly non-trivial (e.g., can contain the free groupC ∗ -algebras as quotients [46]). Nonetheless, such “genuinely quantum” quantum permutationsarise naturally in a variety of contexts. For example, in quantum information theory, quantumpermutation matrices arise naturally in the framework of non-local games and go under the name“projective permutation matrices” [4, 36, 34, 35]. From the perspective of non-commutative geom-etry and quantum group theory, N × N quantum permutation matrices were discovered by Wang[46] to be precisely the structure that encode the quantum symmetries of a finite set of N points.More precisely, Wang considered the universal unital C ∗ -algebra A = C ∗ (cid:16) u ij , ≤ i, j ≤ N (cid:12)(cid:12) u ij = u ∗ ij = u ij & X i u ij = X j u ij = 1 ∀ i, j (cid:17) , generated by the coefficients of a “universal” N × N quantum permutation matrix. Wang thenshowed that there exists a compact quantum group , S + N , acting universally and faithfully on theset of N points in such a way that A gets identified with the C ∗ -algebra C ( S + N ) of “continuousfunctions” on the “quantum space” S + N . The quantum group S + N is called the quantum permutationgroup or quantum symmetry group of N points . In contrast to its classical counterpart, S + N (or moreprecisely A = C ( S + N )) is a highly non-commutative and infinite-dimensional object. It is one of ourmain goals in this paper is to investigate to what extent the quantum permutation groups S + N (andthe quantum reflection groups) can be approximated by elementary finite-dimensional structures.We elaborate briefly on this now.C ∗ -algebraic compact quantum groups as introduced in [48] form the basis of what by nowis a rich theory, developing rapidly in a number of different directions. As indicated above, theperspective we adopt in this paper is that the Hopf (C ∗ -)algebras studied in [48, 49, 50] play the roleof function algebras on a “compact quantum space” G which is equipped with a group structure,and can equivalently be viewed as the complex group algebras of the Pontryagin dual “discretequantum group” Γ. To keep matters simple, throughout this introduction we write C Γ for thegroup algebra of a discrete quantum group Γ (see Section 2 below for details).One aspect of the theory of discrete quantum groups that presents itself naturally from thispoint of view is that of approximation properties . This typically refers to the “accessibility” of thequantum group (or its associated algebras) via finite structures of some type. The phrase ‘finitestructure’ is purposely vague, and lends itself to a variety of interpretations, for example: • The amenability of a discrete quantum group implies the nuclearity of C ∗ (Γ), the universalC ∗ -completion of C Γ. (I.e., the identity map on C ∗ (Γ) can be point-norm approximated byfinite-rank completely positive contractions). The converse is also true, at least for Kac typediscree quantum groups [14]. 2 The
Haagerup approximation property of Γ corresponds to the point-norm approximationof the identity map on the reduced C ∗ -algebra C ∗ r (Γ) by certain well-behaved L -compact,contractive completely positive maps [26]. • The weak amenbility of Γ corresponds to the point-norm approximation of the identity mapon the reduced C ∗ -algebra C ∗ r (Γ) by certain well-behaved finite rank, uniformly boundedcompletely bounded maps.The above are only a few scattered examples, as we cannot possibly do justice to the vast literaturehere. We refer to the survey [21] and its sources for a more expansive discussion on approximationproperties for discrete (in fact locally compact) quantum groups. The above list (and all thoseconsidered in [21]) can be thought of as instances internal approximation of a quantum group byfinite structures, since all the above approximating maps are from a given object to itself. In(quantum) group theory there are of course approximation properties which have an external flavorin the sense that one approximates a given large object by mapping it into smaller auxilliary objects.For example, • A Kac type discrete quantum group Γ is hyperlinear if its quantum group von Neumannalgebra L (Γ) admits-finite dimensional matrix models relative to its Haar trace [22]. • A Kac type discrete quantum group Γ has the
Kirchberg factorization property if there is anet ϕ k : C ∗ (Γ) → M n ( k ) ( C ) of contractive completely positive maps which are asymptoticallytrace-norm multiplicative and satisfy h = lim k tr n ( k ) ◦ ϕ k pointwise, where h is the Haar traceand tr n ( k ) is the normalized matrix trace. • A (finitely generated) discrete quantum group Γ is residually finite if the points of C Γ areseparated by its finite-dimensional ∗ -representations [24, 16]. • A stronger form of residual finiteness of interest for us is the existence of a faithful or innerfaithful matrix model π : C Γ → M N ( C ( X )) for some compact Hausdorff X [9, 11, 12, 7]; wewill have more to say about this concept below.It is the above list of external approximation properties that we are interested in establishingfor (the duals of) Wang’s quantum permutation groups S + N and their generalizations H s + N (theso-called quantum reflection groups [13]). We recall some of the details in the preparatory Section 2below, and for now content ourselves to only remind the reader that H s + N is a quantum version ofthe classical subgroup H sN ⊂ GL N consisting of N × N monomial matrices whose non-zero entriesare s th roots of unity (so in particular H N ∼ = S N , the symmetric group on N symbols).The non-commutative topology of H s + N as a compact quantum group plays a central role in ourstudy of finiteness and approximation properties for their discrete duals d H s + N , and hence the typesof results proven in the paper. We elaborate briefly: Let G be a compact quantum group and let G i < G be a family of closed quantum subgroups of G . The condition that G i topologically generate G was introduced in [22, Definition 4] for a pair of subgroups, but generalizes readily to arbitraryfamilies. In that paper, topological generation is used in the same fashion we do here: as a tool forlifting finiteness properties from the duals Γ i = c G i to Γ = b G . For that reason, we prove a numberof topological generation results (Theorems 3.3 and 3.12) that can be summarized as: Theorem
For all ≤ s ≤ ∞ and N ≥ , the quantum reflection group H s + N is topologicallygenerated by its quantum subgroups S N and H s + N − . For s = 1 the result also holds for N = 5 . (cid:4) N ≥ U + N is topologically generated by its quantum subgroup S × U + N − (product in thecategory of compact quantum groups, dual to the co product C ( S ) ∗ C ( U + N − ) of C ∗ -algebras) andthe classical unitary subgroup U N < U + N . As for residual finiteness results, we use these topologicalgeneration results to prove (see Theorems 3.6 and 3.11): Theorem
For N ≥ and ≤ s ≤ ∞ the discrete duals d H s + N of the quantum reflection groups areresidually finite. (cid:4) Here too there are precedents for other families: [24] treats the discrete duals of free unitary andorthogonal quantum groups.We regard the above theorem as one of the main results of the paper, but it has a number ofpowerful consequences, including the hyperlinearity and Kirchberg factorization property for theselfsame discrete quantum groups [16], as well as improved estimates for the free entropy dimensionof the generators of the associated von Neumann algebras L ∞ ( H s + N ) - see Section 3.4.At this point it is worth highlighting that although our strategy for proving residual finitenessresults for the quantum groups H s + N (by means of inductive topological generation methods) isthe same as that used in prior works, there is one critical difference here. Unlike in the case ofthe free unitary/free orthogonal quantum groups which rely on the existence of “large” smooothLie subgroups (namely U N and O N , respectively), the quantum reflection groups only admit finiteclassical subgroups. This difference turns out to be a fundamental obstruction to a straightforwardextension of the inductive arguments of [24, 22]. To bypass this issue, we make essential use of arecent remarkable result of Banica [5, Theorem 7.10] which establishes that there is no intermediatequantum subgroup for the inclusion S < S +5 . The maximality of the inclusion S N < S + N is widelyconjectured to be true for all N , and the case N = 5 solved by Banica in [5] represents a majoradvancement on this conjecture. It is also interesting to note that Banica’s proof of the maximalityof the inclusion S < S +5 is based on a reduction of this problem to the seemingly different problemof classifying II -subfactors at index 5. This latter problem, however, has recently been solved[28, 27]. The authors find this connection to the classification of subfactors highly intriguing.The other major set of results in this paper pertains to the quantum permutation groups S + N . In this case it turns out that we can say quite a lot more at the level of finite-dimensionalrepresentations.While our residual finiteness results ensure the existence of enough finite-dimensional ∗ -representationsto separate points in the group algebras A ( S + N ) = C c S + N , it is often desirable to have a single rep-resentation π : A ( S + N ) → B , where B is some “nice” C ∗ -algebra (e.g. finite-dimensional, or of theform M N ( C ( X ))) which encodes enough information about S + N so as to generate an asymptoticallyfaithful sequence of finite-dimensional representations ( π k ) k ∈ N of A ( S + N ). The relevant concept weare after here is that of an inner faithful represention π : A ( S + N ) → B . We defer the precise defini-tion of inner faithfulness to Section 4 but note here that a representation π : A ( S + N ) → B is innerfaithful if and only if the sequence of representations π k : A ( S + N ) → B ⊗ k ; π k = π ⊗ k ◦ ∆ ( k ) ;is asymptotically faithful in the sense that T k ker π k = { } [39]. In the above, ∆ : A ( S + N ) →A ( S + N ) ⊗ A ( S + N ) denotes the coproduct and ∆ ( k ) = (id ⊗ ∆ ( k − ) ◦ ∆ for all k ≥
2. In particular, thismeans that A ( S + N ) faithfully embeds into a C ∗ -ultraproduct of the sequence of algebras ( B ⊗ k ) k ∈ N .One particular “minimal ” representation of A ( S + N ) that has been conjectured to be inner faithful4s Banica’s universal flat representation [12, 7]. This particular representation takes the form π : A ( S + N ) → M N ( C ( X N )), where X N ⊂ M N ( M N ( C )) is the compact space of all N × N bistochasticmatrices P = ( P ij ) i,j whose entries are rank-one projections in P ij ∈ M N ( C ). In this paper, weuse modifications of our topological generation results to verify the conjectured inner faithfulnessof the representation π for almost all values of N (cf. Corollary 4.9). Theorem
For all N ≤ and N ≥ , the universal flat matric model π : A ( S + N ) → M N ( C ( X N )) is inner faithful. Piggybacking on the proof of this result, we are able to moreover show that one can reduce thebase space X N to only contain at most 3 points and still achieve an inner faithful finite-dimensionalrepresentation. In other words, we have (cf. Theorem 4.11) Theorem
For all N ≤ and N ≥ , the quantum permutation group algebras A ( S + N ) are innerunitary: they admit inner faithful ∗ -homomorphisms π : A ( S + N ) → B , with B a finite dimensionalC ∗ -algebra. It is an artifact of our proof that we are unable to settle the cases N ∈ [6 ,
9] in the abovetheorems. Our arguments rely heavily on our topological generation results for S + N together withthe crucial result of Banica [5] stating that the inclusion S < S +5 is maximal.The remainder of the paper is organized as follows.Section 2 gathers a number of prerequisites to be used later. In Section 3 we prove some of themain results, Theorems 3.6 and 3.11, to the effect that the discrete Pontryagin duals of the quantumreflection groups are residually finite. This then also implies that they are hyperlinear and have theKirchberg factorization property. The proofs rely in large part on an inductive argument, turningon the fact that quantum permutation groups S + N are topologically generated by their quantumsubgroups S N and S + N − (Theorem 3.3). We also prove a similar result for quantum reflectiongroups in Theorem 3.12; though strictly speaking not needed for residual finiteness of the quantumreflection groups, it might nevertheless be of some independent interest.In the final section 4, we study inner faithful representations of Hopf ∗ -algebras and prove thatthe universal flat representations are inner faithful for N ≤ N ≥
10. In subsection § N the duals A ( S + N ) admit finite-dimensional inner faithful representations. Acknowledgements
M. Brannan and A. Chirvasitu are partially supported by the US National Science Foundation withgrants DMS-1700267 and DMS-1801011 respectively.
Let us start by recalling some facts about compact quantum groups from [50]:
Definition 2.1 A compact quantum group is a unital C ∗ -algebra C ( G ) equipped with a unital ∗ -morphism ∆ : C ( G ) → C ( G ) ⊗ C ( G )5minimal tensor product) such thatspan { ( a ⊗ b ) | a, b ∈ C ( G ) } and span { (1 ⊗ a )∆( b ) | a, b ∈ C ( G ) } are dense in C ( G ) ⊗ C ( G ). (cid:7) As is customary, we regard C ( G ) as the algebra of continuous functions on the fictitious “com-pact quantum space” G . For this reason, the category of compact quantum groups is dual to thatof C ∗ -algebras as in Definition 2.1.For a compact quantum group G we denote the unique dense Hopf ∗ -subalgebra of C ( G ) by A ( G ); it can be regarded alternatively as the complex group algebra C b G of the discrete quan-tum group b G whose Pontryagin dual is G , for which reason we might occasionally revert to thatalternative notation for it. It is often also rendered as Pol( G ) or O ( G ) in the literature.The C*-algebra C ( G ) is equipped with a unique state h : C ( G ) → C (its Haar state ) that is leftand right-invariant in the sense that the diagram C ( G ) C ( G ) ⊗ C ( G ) C ( G ) C ∆ h ⊗ id h and the analogous mirror diagram (obtained by substituting id ⊗ h for the upper right hand arrow)both commute. We will also occasionally refer to the quantum group von Neumann algebra L ∞ ( G ),which is by definition the von Neumann algebra generated by the GNS representation of C ( G )associated to h . Often in the literature the von Neumann algebra L ∞ ( G ) is written as L ( b G )in view of it being a generalization of the von Neumann algebra generated by the left-regularrepresentation of a discrete group. Definition 2.2 A representation of the compact quantum group G is a finite-dimensional comoduleover the Hopf ∗ -algebra A ( G ). We write Rep( G ) for the category of representations of G . (cid:7) We refer to [38] for the relatively small amount of background needed here on comodules overcoalgebras or Hopf algebras.
Definition 2.3
Given two compact quantum groups H and G , we say that H is a (closed) quantumsubgroup of G if there exists a surjective Hopf ∗ -algebra morphism π : A ( G ) → A ( H ). In this case,we write H < G . (cid:7) Note that if H < G as above and V is a G -comodule, then V automatically becomes a H -comodule in a natural way. Indeed if α : V → V ⊗ A ( G ) is the associated corepresentation definingthe comodule V and π : A ( G ) → A ( H ) is the surjective Hopf ∗ -morphism from above, then V becomes an H -comodule via the corepresentation (id ⊗ π ) α : V → V ⊗ A ( H ). This “restriction” ofrepresentations of G to representations of H , induces, at the level of Hom-spaces, natural inclusionshom G ( V, W ) ֒ → hom H ( V, W ) , for any pair of G comodules V, W . 6 .2 Quantum reflection groups
Quantum reflection groups were introduced in [13, Definition 1.3] based on earlier work done in [6].Following the former reference we denote them by H s + N . Definition 2.4
Let N ≥ ≤ s ≤ ∞ .The underlying Hopf ∗ -algebra A = A ( H s + N ) of the quantum reflection group H s + N is generatedas a ∗ -algebra by N normal elements u ij , 1 ≤ i, j ≤ N such that • the matrices u = ( u ij ) i,j and u t = ( u ji ) i,j are both unitary in M N ( A ); • p ij = u ij u ∗ ij is a self-adjoint idempotent for each 1 ≤ i, j ≤ N ; • u sij = p ij for each 1 ≤ i, j ≤ N ,with the last relation absent when s = ∞ .The coproduct ∆ : A → A ⊗ A is determined by∆( u ij ) = N X k =1 u ik ⊗ u kj (1 ≤ i, j ≤ N ) . (cid:7) The quantum groups H s + N are meant to be quantum analogues of their classical versions H sN consisting of monomial N × N matrices whose non-zero entries are s th roots of unity. In particular, s = 1 recovers the symmetric group S N ; this is also the case in the quantum setting, recovering thequantum groups introduced in [46]: Definition 2.5
Let N ≥
2. The quantum symmetric group S + N is the quantum reflection group H N from Definition 2.4. Its Hopf ∗ -algebra A is freely generated as a ∗ -algebra by an N × N magic unitary matrix u = ( u ij ) ij ∈ M N ( A ), in the sense that all of the entries of u are projectionsand the entries from each row and column add up to 1 ∈ A . (cid:7) Now let N ≥ ≤ s, t < ∞ be positive integers, as in Definition 2.4. The generators u ij of A ( H s + N ) clearly satisfy the defining relations of A ( H st + N ) as well, so we get a surjective Hopf ∗ -algebra morphism A ( H st + N ) ∋ u ij u ij ∈ A ( H s + N ) . In other words, we have natural embeddings H s + N < H st + N . These embeddings will feature againbelow.Also for future use, we briefly recall some background on the representation theory of thequantum groups H s + N from [13]. Let F s be the free monoid on the symbols in Z s (understood as Z when s = ∞ ), equipped with the following operations: • The involution x x defined by a · · · a k ( − a k ) · · · ( − a ) , a i ∈ Z s ; • The fusion binary operation ‘ · ’ defined by( a · · · a k ) · ( b · · · b ℓ ) = a · · · a k − ( a k + b ) b · · · b ℓ . R s of the category of finite-dimensional representations of H s + N has a basis { x f } over Z indexed by f ∈ F s , and the multipli-cation resulting from the tensor product x f x g = X f = vz,g = zw ( x vw + x v · w ) . We remark that [13, Theorem 7.3] is not clearly stated in this manner for s = ∞ , but the proofgoes through essentially unchanged for both finite and infinite s . A more general argument can befound in [32].Restriction via the inclusion H s + N < H st + N induces a ring morphism R st → R s sending thegenerator x , 1 ∈ Z st to x , 1 ∈ Z s . In particular, we have Lemma 2.6
The natural map R ∞ → lim ←− s R s is an embedding, where the directed inverse limit is taken over the positive integers ordered bydivision: s ≤ st . (cid:4) The following is a combination of [22, Definition 2] and [16, Definition 1.12].
Definition 2.7
A discrete quantum group b G is finitely generated if C b G is finitely generated as analgebra. If b G is finitely generated then we say it is residually finite if C b G embeds as a ∗ -algebrainto a (possibly infinite) direct product of matrix algebras.Moreover, b G is said to have the Connes embedding property (or is
Connes-embeddable or hyper-linear ) if it is of Kac type and the von Neumann algebra ( L ∞ ( G ) admits a Haar state-preservingembedding into the ultrapower R ω of the hyperfinite II -factor. (cid:7) Remark 2.8
Note that residual finiteness implies the
Kirchberg factorization property of [17, Def-inition 2.10] by [16, Theorem 2.1]. In turn, Kirchberg factorization implies hyperlinearity (e.g. [16,Remark 2.6]); in conclusion, residual finiteness is stronger than Connes embeddability for a discretequantum group. (cid:7)
We also recall from [22, Definition 4] the notion of topological generation for compact quantumgroups:
Definition 2.9
A family of compact quantum subgroups ( G i < G ) i ∈ I topologically generate G if,for every pair of representations V , W of G , the natural inclusion maphom G ( V, W ) ֒ → \ i ∈ I hom G i ( V, W )is an isomorphism. In this case, we write G = h G i i i ∈ I (cid:7) We will need the following alternative description of topological generation, which is almostimmediate given Definition 2.9; see also [22, Proposition 3.5].
Lemma 2.10
A family of quantum subgroups ( G i < G ) i ∈ I topologically generates G if and only iffor every G -representation V a map f : V → C that is a morphism over every G i is a morphismover G (i.e., f ∈ ∩ i ∈ I hom G i ( V, C ) = ⇒ f ∈ hom G ( V, C ) ). Equivalently, it suffices to check this forall irreducible representations V . (cid:4)
8n particular, we have the following sufficient criterion for topological generation:
Corollary 2.11
Let ( G i < G ) i ∈ I be a family of quantum subgroups of a compact quantum groupand assume that for every irreducible non-trivial G -representation V there is some i such that therestriction of V to G i contains no trivial summands.Then, G is topologically generated by the G i . Proof
This follows from Lemma 2.10: a morphism f : V → C of representations witnesses anembedding of the trivial representation into V , and the hypothesis ensures that a non-trivial ir-reducible V ∈ Rep( G ) retains the property of having no trivial summands over G i for some i ,meaning that \ i ∈ I hom G i ( V, C ) = { } = hom G ( V, C ) . (cid:4) Topological generation appears under a different name in [24, § Lemma 2.12
A finitely generated discrete quantum group b G is residually finite if and only if itsdual G is topologically generated by a family of subgroups G i < G with residually finite c G i . (cid:4) Recall the embeddings H s + N < H st + N from § Lemma 2.13
For N ≥ the quantum group H ∞ + N is topologically generated by its quantum sub-groups H s + N for finite s . Proof
According to Lemma 2.6 the criterion of Corollary 2.11 is satisfied: indeed, the former resultensures that every non-trivial irreducible H ∞ + N -representation remains irreducible and non-trivialover some H s + N . (cid:4) We focus here on ∗ -algebras satisfying the condition required of C b G in Definition 2.7: Definition 2.14 A ∗ -algebra A is residually finite-dimensional or RFD if it embeds as a ∗ -algebrain a product of matrix algebras. (cid:7) Remark 2.15
The notion is used frequently in the context of C ∗ -algebras, but here we are inter-ested in the purely ∗ -algebraic version. (cid:7) We gather a number of general observations on the RFD property for later use. First, sinceresidual finite-dimensionality obviously passes to ∗ -subalgebras, we will need to know that certainnatural morphisms between free products with amalgamation (or pushouts , as we will also referto them) are embeddings. The following result is likely well known, but we include it here forcompleteness. Lemma 2.16
Suppose we have the following commutative diagram of complex algebras, all of whosearrows are embeddings.
D AB D A ′ B ′∼ = If D is finite-dimensional and semisimple then the canonical map A ∗ D B → A ′ ∗ D B ′ is one-to-one. roof Denote by D ◦ the opposite algebra of D . Under the hypothesis on D the enveloping algebra D ⊗ D ◦ is semisimple, and hence the category of D -bimodules (which are simply D ⊗ D ◦ -modules)is semisimple. It follows that the inclusions D → A → A ′ are split as D -bimodule maps, and hence we have direct sum decompositions A = D ⊕ A , A ′ = D ⊕ A ′ = D ⊕ A ⊕ A (1)and similarly for the B side of the diagram.According to [15, Corollary 8.1] the pushout A ∗ D B decomposes as a direct sum of tensorproducts of the form T ⊗ T · · · ⊗ T k , T i chosen alternately from {A , B } (2)in the category of D -bimodules. The analogous decomposition holds for A ′ ∗ D B ′ , and due to (1)the tensor product (2) is a summand in its counterpart T ′ ⊗ · · · ⊗ T ′ k , T ′ i ∈ {A ′ , B ′ } alternately , T ′ i = A ′ ⇐⇒ T i = A . The conclusion that
A ∗ D B → A ′ ∗ D B ′ is an embedding follows. (cid:4) There are analogues of this in the C ∗ -algebra literature: see e.g. [3, Proposition 2.2] and [37,Theorem 4.2]. We will only use Lemma 2.16 in the context of ∗ -algebras, with all embeddings being ∗ -algebra morphisms. An immediate consequence of Lemma 2.16 is Corollary 2.17
Under the hypotheses of Lemma 2.16, suppose furthermore that all maps are ∗ -algebra morphisms. If A ′ ∗ D B ′ is RFD, then so is A ∗ D B . (cid:4) We now turn to the technical result of this section, which will be needed later on.
Proposition 2.18 If A is an RFD ∗ -algebra and D ⊂ A is a finite-dimensional commutativeC ∗ -subalgebra then A ∗ D A is RFD. Before going into the proof we treat a particular case.
Lemma 2.19
Proposition 2.18 holds when A is a finite-dimensional C ∗ -algebra. This in turn requires some preparation. More precisely, we first build a faithful Hilbert spacerepresentation of the pushout A∗ D A . Throughout the present discussion D ⊂ A are as in the state-ment of Lemma 2.19. Consider the canonical conditional expectation E : A → D that preserves anarbitrary but fixed faithful tracial state τ on A . This means that A D C E τ | D τ Moreover, the map E is contractive, completely positive, and splits the inclusion D → A in thecategory of D -bimodules (see e.g. [40, § IX.4] for background on expectations on operator subalge-bras). 10e denote by A ℓ and A r the copies of ker( E ) in the left and respectively right hand side freefactor of A ∗ D A . Then, as a consequence of [15, Corollary 8.1], we have A ∗ D A = M i T i where i ranges over the words on ℓ, r with no consecutive repeating letters and e.g. T ℓrℓ ··· = A ℓ ⊗ A r ⊗ A ℓ ⊗ · · · with tensor products over D . The empty word is allowed, with T ∅ = D .Each T i is then naturally a Hilbert D -module (e.g. [47, Chapter 15]). Composing the D -valuedinner product on T i further with the inner product h x | y i = τ ( x ∗ y ) , x, y ∈ D makes each T i into a Hilbert space. Left multiplication makes the algebraic direct sum L i T i afaithful module over A ∗ D A (it is simply the left regular representation of the algebra in question).The Hilbert space completion of L i T i is thus a faithful Hilbert space representation of A ∗ D A . Proof of Lemma 2.19
The fact that the full C ∗ -pushout A ∗ D A is residually finite-dimensionalfollows from [3, Theorem 4.2]. It thus remains to show that the algebraic pushout A ∗ D A embedsin its C ∗ -envelope.In other words, we want to argue that A ∗ D A admits a faithful representation on some Hilbertspace. This, however, is precisely what we constructed above in the discussion preceding the proof:the Hilbert space direct sum of T i for i ranging over words on { ℓ, r } with no repeating consecutiveletters is such a representation. (cid:4) Remark 2.20
The construction of the Hilbert modules T i sketched above features prominently infree probability; see e.g. [43, §
5] and [42, § (cid:7) Proof of Proposition 2.18
The RFD condition implies that there is an embedding
A → Q i ∈ I M n i for an index set I (or arbitrary cardinality). By Corollary 2.17, it thus suffices to assume that A is such a product of matrix algebras to begin with. We therefore make this assumption throughoutthe rest of the proof.We first appeal once more to [15, Corollary 8.1] to conclude that since the embedding D → A splits as A = D ⊕ A in the category of D -bimodules (because D ⊗ D ◦ is semisimple), we have a decomposition A ∗ D A ∼ = M T ⊗ · · · ⊗ T k (3)where T i are chosen from among the two copies of A . An arbitrary element in A ∗ D A can thusbe expressed as a sum of elements of finitely many tensor products as in (3). It follows that if x isnon-zero then it maps to a non-zero element of a pushout B ∗ D B through some projection A = Y i ∈ I M n i → Y i ∈ F M n i = B for a finite subset F ⊆ I . In conclusion, it will be enough to further assume that A is a finiteproduct of matrix algebras, i.e. a finite-dimensional C ∗ -algebra. This is Lemma 2.19, hence theconclusion. (cid:4) emark 2.21 Once more, there are versions of Proposition 2.18 applicable to C ∗ -algebras; [33,Corollary 2] is one example. (cid:7) We conclude this section with one more proposition which will have direct application in thenext section to the residual finiteness of the duals of quantum reflection groups.
Proposition 2.22
Let B be an RFD ∗ -algebra, D ⊂ B a finite-dimensional commutative C ∗ -subalgebra, and C a finite-dimensional C ∗ -algebra. Then, C ∗ B / [ C , D ] is RFD. Proof
The algebra in question is isomorphic to the pushout(
C ⊗ D ) ∗ D B . Since we have rightward embeddings
D C ⊗ DB D C ⊗ BC ⊗ B ∼ = Corollary 2.17 shows that it will be enough to prove residual finite-dimensionality for(
C ⊗ B ) ∗ D ( C ⊗ B )In turn, this follows from Proposition 2.18 and the simple observation that tensor products of RFD ∗ -algebras are RFD. (cid:4) Our first goal in this section is to treat the case of quantum permutation groups. Before getting tothe main results, we first recall some basic facts about the description of the invariant theory for S + N in terms of non-crossing partitions [10]. Definition 3.1
Fix k ∈ N and consider the ordered set [ k ] = { , . . . , k } . A partition of [ k ] is adecomposition p of [ k ] into a disjoint union of non-empty subsets, called the blocks of p . A partition p of [ k ] has a crossing if there exist a < b < c < d ∈ [ k ] such that { a, c } and { b, d } belong todifferent blocks. A partition p is called non-crossing if it has no crossings. The collection of allnon-crossing partions of [ k ] is denoted by N C ( k ). (cid:7) Given a function i : [ k ] → [ N ] (i.e. a multi-index i = ( i (1) , . . . , i ( k )) ∈ [ N ] k ), we let ker i be thepartition of [ k ] given by declaring that r, s belong to the same block of ker i if and only if i ( r ) = i ( s ).Given i as above and p ∈ N C ( k ), we define δ p ( i ) = ( i ≥ p , ≥ denotes the refinement partial order on the lattice partitions of [ k ].Now let V be an N -dimensional Hilbert space with distinguished orthonormal basis e j , 1 ≤ j ≤ N . Given k, l ∈ N and p ∈ N C ( k + l ), we form the linear map T k,l,Np : V ⊗ k → V ⊗ l ; T k,l,Np ( e i (1) ⊗ . . . ⊗ e i ( k ) ) = X j :[ l ] → [ N ] δ p ( ij ) e j (1) ⊗ . . . ⊗ e j ( l ) , where ij : [ k + l ] → [ N ] is the concatenation of i and j . We call such linear maps non-crossingpartition maps . The fundamental result that will be of use to us here is the following descriptionof S + N invariants in terms of these non-crossing partition maps. Theorem 3.2 ([10])
For each k, l ∈ N , N ∈ N we have linear isomorphismshom S + N ( V ⊗ k , V ⊗ l ) = span { T k,l,Np : p ∈ N C ( k, l ) } . Moreover, if N ≥ , then { T k,l,Np : p ∈ N C ( k, l ) } forms a linear basis for hom S + N ( V ⊗ k , V ⊗ l ) . Let V = span( e i ) Ni =1 be as above, regarded as the fundamental representation of S + N . We regard S + N − as also acting on V via its fundamental representation on the ( N − V N spanned by e i , 1 ≤ i ≤ N −
1, and via the trivial representation on the orthogonal complement C e N . In this way, we regard S + N − < S + N . For future reference, we denote by V i the subspace of V spanned by all e j , j = i , and by W i the span of e i alone.Our first main result in this section reads as follows. Theorem 3.3
For N ≥ the quantum permutation group S + N is topologically generated by itssubgroups S N and S + N − . Before embarking on the proof we record the following immediate strengthening of the state-ment:
Corollary 3.4
Let N ≥ , ≤ M ≤ N and S + M < S + N the embedding of the quantum subgroupfixing N − M of the standard basis vectors of the defining representation of S + N on C N . Then, S + N is topologically generated by S N and S + M . Proof
This is a repeated application of Theorem 3.3: S M < S N and S + M topologically generate S + M +1 and the result follows by induction. (cid:4) The cases N = 5 and N ≥ V is the N -dimensional Hilbert space with distinguished orthonormal basis e j , 1 ≤ j ≤ N carrying the defining representation of S + N . Proposition 3.5
Suppose N ≥ . For any k ≥ , any linear map f : V ⊗ k → C that is invariantunder both S + N − and the classical permutation group S N is invariant under all of S + N . Proof
In other words, we have to show a functional f : V ⊗ k → C that is both an S + N − -coinvariantand an S N -coinvariant must also be an S + N -coinvariant.Note that the S N -invariance ensures that f respects the action of each one among the N − S + N − < S + N obtained by acting on the subspaces V i ⊂ V . Moreover,13he invariance under the original copy of S + N − < S + N ensures that when restricted to V ⊗ kN , f acts as a linear combination of the non-crossing partition maps { T k, ,N − p } p ∈ NC ( k ) . Replacingthe maps { T k, ,N − p } p ∈ NC ( k ) in this linear combination with { T k, ,Np } p ∈ NC ( k ) (which belong tohom S + N ( V ⊗ k , C )) and subtracting this new linear combination from f , we may as well assume that f | V ⊗ kN vanishes and then try to prove that f itself is zero.Once more, the S N -invariance ensures that the restriction of f to V ⊗ ki ⊂ V ⊗ k vanishes for every1 ≤ i ≤ N . What we have to show, however, is that it also vanishes on summands of V ⊗ k obtainedby tensoring some copies of V N with some copies of W N . To simplify notation and fix ideas, wewill show that the restriction of f to, say, U = V ⊗ ( k − l ) N ⊗ W ⊗ lN ⊂ V vanishes. The general case is perfectly analogous, with only notational difficulties making thepresentation more cumbersome.As the action of the original copy of S + N − that we considered is trivial on W N , U can beidentified with the ( k − l ) th tensor power of the fundamental representation of S N − , and henceany S + N − -coinvariant U → C will be some linear combination of non-crossing partitions mapsassociated to N C ( k − l ).Now let V ,N ⊂ V N be the span of e j , j = 1 , N . Because N ≥ V ,N is at least 4-dimensionaland hence the linear forms associated to non-crossing partitions are linearly independent on it. Thismeans that if a linear combination of non-crossing partition functionals on V ⊗ ( k − l )1 ,N → C vanishes,then the linear combination itself must be trivial. But note now that f restricted to V ⊗ ( k − l )1 ,N ⊗ W ⊗ lN vanishes, because the space in question is a subspace of V ⊗ k . By the paragraph above, f musttherefore vanish on all of U . (cid:4) Proof of Theorem 3.3
As mentioned above, we treat the cases N = 5 and N > (Case 1: N ≥ ) Recall e.g. from [8, Theorem 4.1] that every finite-dimensional S + N -representation appears as a summand of V ⊗ k where V is the N -dimensional defining represen-tation and k is some positive integer. By Lemma 2.10 the conclusion is now a paraphrase ofProposition 3.5. (Case 2: N = 5 ) According to [5, Theorem 7.10] the inclusion S < S +5 admits no intermediatequantum groups. Since S +4 < S +5 is not a quantum subgroup of S , we indeed have S +5 = h S , S +4 i . (cid:4) As a consequence of the above we have
Theorem 3.6
The discrete duals c S + N of the free quantum permutation groups are residually finite. Proof
By Lemma 2.12 and theorem 3.3 we can proceed inductively once we know that • c S N is residually finite (the group algebra is finite-dimensional); • c S +4 is residually finite.For the latter, recall from [11, Definition 2.1 and Theorem 4.1] that A ( S +4 ) embeds into the C ∗ -algebra C ( SU , M ), and hence has enough 4-dimensional representations. (cid:4) Since, as observed in Remark 2.8, hyperlinearity and the Kirchberg factorization property areweaker than residual finiteness, we also have 14 orollary 3.7
The discrete duals c S + N have the Kirchberg factorization property and are hyperlin-ear. (cid:4) It will be of some interest to have alternative topological generation results which we now stateand prove. Let 4 ≤ M ≤ N be a pair of positive integers, and write N = M + T . We then haveHopf ∗ -algebra surjections A ( S + N ) → A ( S + M ) ∗ A ( S + T )that annihilate u ij for i, j in distinct parts of any partition of [ N ] into two parts of sizes M and T . We will be somewhat vague on which partitions to use; sometimes we need to refer to arbitraryones, but when we do not, the reader can simply assume the partition is[ N ] = [ M ] ⊔ { M + 1 , · · · , M + T } , (4)corresponding to the upper left-hand corner embedding S + M < S + N corresponding to action of S + M on V = span { e , . . . , e N } which fixes e M +1 , . . . , e M + T .We now come to a critical notion. Definition 3.8
An compact quantum group embedding G < S + N is ( M, N ) -large if it factors theupper path in the diagram A ( S + N ) A ( S + M ) ∗ A ( S + T ) A ( G ) A ( S + M ) (5)so as to make its lower path one-to-one. (cid:7) Example 3.9
The obvious examples of (
M, N )-large embeddings are those corresponding to thestandard surjections A ( S + N ) → A ( S + M ). Slightly less obvious examples can be obtained as follows:Suppose N = KM for some positive integer K . The diagonal embedding S + M ≤ S + N is obtained atthe level of Hopf algebras as the surjection A ( S + N ) → A ( S + M ) ∗ K → A ( S + M )where the left hand arrow annihilates the generators u ij that are off the diagonal consisting of M × M blocks and the right hand arrow is the identity on each free factor. Diagonal embeddingsare ( M, N )-large in the sense of Definition 3.8. (cid:7)
The embedding of Hopf ∗ -algebras A ( S + M ) → A ( G ) forming the lower half of (5) correspondsto a morphism of quantum groups G → S + M , which in turn gives rise to restriction and inductionfunctors Rep( S + M ) Rep( G ) resind (6)with restriction being the left adjoint to induction. These remarks will recur below. Proposition 3.10
Let ≤ M ≤ N and G < S + N an ( M, N ) -large embedding. Then, S + N istopologically generated by G and S N . roof Throughout the proof we will fix a basis { e i } for the N -dimensional carrier space V of thedefining representation of S + N , and assume that the upper left hand arrow in (5) is the standard onecorresponding to the partition (4). The proof will be very similar to the that of Proposition 3.5:we fix a map f : V ⊗ k → C that is a morphism both over S N < S + N and G and seek to show that f is also an S + N -morphism.Decompose V = V ⊕ V where V = span { e , · · · , e M } , V = span { e M +1 , · · · , e M + T } . The tensor power V ⊗ k then decomposes as M i V i ⊗ · · · ⊗ V i k with summands ranging over all tuples i = ( i j ), i j ∈ { , } . The summand V ⊗ k is a comoduleover A ( S + M ) ⊂ A ( G ) (the embedding being the lower path in (5)), and hence f | V ⊗ k is a linearcombination of non-crossing partitions. Subtracting the same linear combination of non-crossingpartitions on V ⊗ k , we can assume f | V ⊗ k = 0. The goal now is to show that in fact f = 0 globally(i.e. on the entirety of V ⊗ k ). We do this iteratively, proving by the following t -dependent claim byinduction on t : Claim( t ): The restriction of f to any summand of the type V i ⊗ · · · ⊗ V i k , t of the i j are 2 . (7)is zero.The base case t = 0 of the induction is in place (claiming simply that f | V ⊗ k = 0, which weknow). We now turn to the induction step, assuming Claim( s ) for all s ≤ t − t ). In order to lessen the notational load of the argument we will focus on V ⊗ ( k − t )1 ⊗ V ⊗ t (i.e. we assume the last t indices in (7) are 2).Equivalently, we have to show that f is zero on V ⊗ ( k − t )1 ⊗ C e j ⊗ V ⊗ ( t − for any j ∈ [ M + 1 , M + T ]. To do this, first note that because V is self-dual over S + M ∗ S + T , hencealso over its quantum subgroup G , the G -morphisms V ⊗ ( k − t )1 ⊗ V ⊗ t → C are in bijection with the G -morphisms V ⊗ ( k − t )1 → V ⊗ t . Since moreover V is the induction of the corresponding S + M -representation, the adjunction (6) readshom G (cid:16) V ⊗ ( k − t )1 , V ⊗ t (cid:17) ∼ = hom S + M (cid:16) V ⊗ ( k − t )1 , ind V ⊗ t (cid:17) (8)Now, the finite-dimensional S + M -representation ind V ⊗ t embeds into some tensor power V ⊗ s , andhence elements in the right hand side of (8) are spanned by non-crossing partitions of [ s + k − t ].Since non-crossing partitions are linearly independent on spaces of dimension ≥
4, morphisms in(8) vanish if they do so when restricted to( V ′ ) ⊗ ( k − t ) , V ′ = span { e , · · · , e M − } .
16n our setting, what this means is that it is enough to prove that f vanishes on( V ′ ) ⊗ ( k − t ) ⊗ C e j ⊗ V ⊗ ( t − . (9)Set V ′ = span { e ℓ | ℓ ∈ [ M + 1 , M + T ] − { j }} . Then, ( V ′ ) ⊗ ( k − t ) ⊗ C e j ⊗ ( V ′ ) ⊗ ( t − is contained in the image of V ⊗ ( k − t +1)1 ⊗ V ⊗ ( t − through athe permutation of the e i interchanging e M and e j . Since f is an S N -morphism, Claim( t −
1) nowimplies that f vanishes on ( V ′ ) ⊗ ( k − t ) ⊗ C e j ⊗ ( V ′ ) ⊗ ( t − . Similarly, f vanishes on the other summands of (9) resulting from the decomposition V = V ′ ⊕ C e j by the other instances Claim( s ), s ≤ t − (cid:4) We now turn to the quantum reflection groups H s + N for 1 ≤ s ≤ ∞ . The main result of thesubsection is Theorem 3.11
For N ≥ and ≤ s ≤ ∞ the dual d H s + N is residually finite and hence alsohyperlinear. Proof
We fix N throughout, and denote by A s and A the Hopf ∗ -algebras associated to H s + N and S + N respectively. (Case 1: s < ∞ ) Recall from [13, Theorem 3.4 (2)] that we have a free wreath productdecomposition A s ∼ = C ( Z s ) ∗ w A , (10)where the right hand side is by definition the free ∗ -algebra generated by A and n copies of C ( Z s ) with the constraint that the i th copy of C ( Z s ) commutes with the i th row of generators u ij ,1 ≤ j ≤ N of A . It follows that A s can be realized as a succession of extensions of the form B 7→ C ( Z s ) ∗ B / [ C ( Z s ) , D ]for a commutative finite-dimensional ∗ -subalgebra D ⊂ B . More concretely, the various algebras D are the N -dimensional subalgebras of A generated by a row u ij , 1 ≤ j ≤ N of generators.The residual finite-dimensionality of A s then follows inductively from Proposition 2.22. (Case 2: s = ∞ ) Given that H ∞ + N is topologically generated by all of the finite H s + N embeddedtherein (by Lemma 2.13), the conclusion follows from Lemma 2.12. (cid:4) While the proof given above for Theorem 3.11 does not proceed inductively on N or requiretopological generation, there is nevertheless an analogue of Theorem 3.3 for quantum reflectiongroups that we record here. Just as we did for quantum permutation groups, we regard H s + N − asa quantum subgroup of H s + N via the map A ( H s + N ) → A ( H s + N − )that sends the generators u ij to δ ij if N ∈ { i, j } . Theorem 3.12
For N ≥ and ≤ s ≤ ∞ the quantum reflection group H s + N is topologicallygenerated by its quantum subgroups H s + N − and S N . roof We once more separate the finite and infinite- s cases. (1: finite s ) This follows from a slight adaptation of Proposition 3.5, modified as followsbased on the representation theory of H s + N as developed in [13, Sections 5-7]. Instead of a single n -dimensional fundamental representation V we have s of them, labeled V ( i ) for i ∈ Z s . As formorphisms, f : V ( i ) ⊗ · · · ⊗ V ( i k ) → C is H s + N -invariant precisely when it is a linear combination of non-crossing partitions whose blocksare of the form { i a , · · · , i a t } , X j i a j = 0 ∈ Z s . The argument in the proof of Proposition 3.5 then goes through virtually unchanged. (2: s = ∞ ) This follows as in the proof of Theorem 3.11, from the claim for s < ∞ and thefact that H s + N topologically generate H ∞ + N (by Lemma 2.13). (cid:4) Remark 3.13
Note the bound on N : unlike Theorem 3.3, Theorem 3.12 does not apply to N = 5hence does not yield an inductive proof of residual finite-dimensionality; we do not know whetherthe result is still valid in that case, but believe it to be. (cid:7) In this section we make some brief remarks on what is currently known about the free entropydimension of the canonical generators of the finite von Neumann algebras L ∞ ( H s + N ). We refer thereader to the survey [44] and the references therein for details on the various versions of free entropydimension that exist.Fix N ∈ N and s ∈ N ∪ {∞} and consider the self-adjoint family X ( N, s ) = { u ij , u ∗ ij } ≤ i,j ≤ N of generators of A ( H s + N ) ⊂ L ∞ ( H s + N ). Associated to the sets X ( N, s ) we have the (modified) microstates free entropy dimension δ ( X ( N, s )) ∈ [0 , n ] and the non-microstates free entropy di-mension δ ∗ ( X ( N, s )) ∈ [0 , n ]. From [18] it is known that the general inquality δ ( · ) ≤ δ ∗ ( · ) alwaysholds, and from [25] an upper bound for δ ∗ exists in terms of the L -Betti numbers of the discretedual quantum groups d H s + N : δ ∗ ( X ( N, s )) ≤ β (2)1 ( d H s + N ) − β (2)0 ( d H s + N ) + 1 . Here β (2) k ( · ) is the k th L -Betti number of a discerete quantum group. See for example [41, 31, 19].Now, in [31, Theorem 5.2], we have the following computations β (2)1 ( d H s + N ) = 1 − s & β (2)0 ( d H s + N ) = 0 ( N ≥ . Finally, since L ∞ ( H s + N ) is Connes embeddable by Theorem 3.11, it follows from [29, Corollary 4.7]that δ ( X ( N, s )) ≥ L ∞ ( H s + N ) is diffuse. The question of when exactly L ∞ ( H s + N ) isdiffuse still seems to be open in complete generality. However, it is known that L ∞ ( H s + N ) is a II -factor (and in particular diffuse) N ≥ Corollary 3.14
For N ≥ and s ∈ N ∪ {∞} , we have ≤ δ ( X ( N, s )) ≤ δ ∗ ( X ( N, s )) ≤ − s . In particular, the generators X ( N ) = { u ij } ≤ i,j ≤ N of L ∞ ( S + N ) satisfy δ ( X ( N )) = δ ∗ ( X ( N )) = 1 for N ≥ . emark 3.15 The situation for S + N in Corollary 3.14 is similar to what happens for the free or-thogonal quantum groups O + N [22]. However, for O + N , even more is known: In [23] it was shownthat in fact L ∞ ( O + N ) is a strongly 1-bounded von Neumann algebra for all N ≥
3. The notion ofstrong 1-boundedness was introduced by Jung in [30] and entails that δ ( X ) ≤ X ⊂ L ∞ ( O + N ). In particular, it follows that L ∞ ( O + N ) is never isomorphic to an in-terpolated free group factor. In this context, it is natural to ask whether similar results hold forquantum permuation groups: Is L ∞ ( S + N ) a strongly 1-bounded von Neumann algebra for all N ≥ ? (cid:7) Theorem 3.6 shows that the quantum group algebras A ( S + N ) = C c S + N have enough finite-dimensional ∗ -representations, i.e. map faithfully into a product of matrix algebras. In the present section weprove that a specific, canonical collection of “elementary” representations is faithful in a certainsense. First, let us clarify the appropriate notion of faithfulness here. See e.g. [9, Definition 2.7]for more details. Definition 4.1 A ∗ -homomorphism π : A → B from a Hopf ∗ -algebra A into a ∗ -algebra B is inner faithful if ker π contains no non-trivial Hopf ∗ -ideals. Equivalently, for any factorization π = ˜ π ◦ ρ with ρ : A → A ′ , a surjective morphism of Hopf ∗ -algebras, we have in fact that ρ isan isomorphism. More generally, the Hopf image of π : A → B is the “smallest” quotient Hopf ∗ -algeba A ′ such that π factors through the quotient map ρ : A → A ′ . (cid:7) Note that the Hopf image always exists and is unique (up to isomorphism) [9]. In this paper,we will only be concerned with the cases where our Hopf ∗ -algebras are of the form A ( G ) for acompact quantum group G . In this case, the Hopf image of π : A ( G ) → B is A ( H ), where H < G is the “smallest” quantum subgroup such that π factors through ρ : A ( G ) → A ( H ). Moreover, π isinner faithful if and only if H = G (up to isomorphism). Following [12, Definition 5.1], we denote by X N the space of N × N matrices P = ( P ij ) ij ∈ M N ( M N ( C )), whose entries P ij ∈ M N ( C ), are rank-one projections with the property that for all1 ≤ i, j ≤ N , X j P ij = 1 = X i P ij In other words, X N ⊂ M N ( M N ( C )) is the compact set of bistochastic N × N matrices of rank oneprojections in M N .It is clear that each P ∈ X N gives rise to a ∗ -homomorphism π P : A ( S + N ) → M N ( C ); π P ( u ij ) = P ij (1 ≤ i, j ≤ N ) , and we call π P a flat matrix model for the quantum group S + N . If we package all these flat matrixmodels π P into one single representation by allowing P ∈ X N to vary, we arrive at a constructionthat features prominently in [12, 7]. Definition 4.2
The universal flat matrix model of S + N is the morphism π : A ( S + N ) → M N ( C ( X N )) ∼ = C ( X N , M N ( C )); π ( u ij ) = { P π P ( u ij ) = P ij } . (11) (cid:7) Conjecture 4.3 [12, Conjecture 5.7] The universal flat matrix model A ( S + N ) → M N ( C ( X N )) isfaithful for N = 4 and inner faithful for N ≥ . Remark 4.4
Note that for N = 4 the conjecture is true by [11, Theorem 4.1] (or [12, Proposition2.1]). Indeed, the latter result produces some faithful representation of the form A ( S +4 ) → M ( C ( X ))for compact X (in fact X = SU ), which must factor as A ( S +4 ) → M ( C ( X )) → M ( C ( X ))by universality. (cid:7) The main result of this section is that this conjecture is true for at least almost all N . Tobegin, we first need a few remarks and observations. Consider the closed subspace X classN ⊂ X N of matrices P ∈ X N for which the entries P ij pairwise commute. Then, the classical permutationgroup S N also has a universal flat matrix model π class : A ( S N ) → M N ( C ( X classN )). Moreover, if q : A ( S + N ) → A ( S N ) is the canonical quotient map and r : C ( X N ) → C ( X classN ) is the restrictionmap, then by construction π class ◦ q = r ◦ π . Lemma 4.5 If N ≥ then the inclusion X classN ⊂ X N is proper. Proof
To produce examples of N × N bistochastic matrices whose entries do not all commute weproceed as follows.First, fix a basis e i , 1 ≤ i ≤ N for C N and let L ∈ M N ( N ) be a Latin square of size N × N (meaning that each row and column is a permutation of { , · · · , N } ). Assume furthermore thatthe upper left hand 2 × L is (cid:18) (cid:19) (12)We can then form the bistochastic and commutative matrix whose ( i, j ) entry is the projection onthe one-dimensional span of e L ij , and then modify it slightly by changing its upper left hand 2 × (cid:18) P u P v P v P u (cid:19) where u = e + e and v = e − e . The resulting bistochastic matrix contains, say, the projections P u and P e , which do not commute.It remains to argue that a Latin square L as above exists if N ≥ N ≥ × n Latin rectangle , in the sense that the two rows are permutations of { , · · · , N } and no twoentries in the same column coincide. We can then use the result that any Latin rectangle can becompleted to a Latin square (e.g. [1, Chapter 35, Lemma 1]). (cid:4) This already yields one instance of Conjecture 4.3.
Proposition 4.6 S +5 satisfies Conjecture 4.3. roof Let π, π class , r, q be as above and assume N = 5. Let I be a Hopf ∗ -ideal contained in ker( π )and observe that π class ( q ( I )) = r ◦ π ( I ) = 0. Since π class is inner faithful (see e.g., [7, Proposition5.3]), this forces q ( I ) = (0), i.e. I ⊂ ker( q ). But because there is no intermediate quantum groupbetween S and S +5 by [5, Theorem 7.10], it follows that either I = (0) or I = ker( q ). In the secondcase we get π = π class ◦ q = r ◦ π. In particular, this implies that any family ( P ij ) i,j N of rank-one projections which are pairwiseorthogonal on rows and columns commutes, i.e. X N = X classN . This equality, however, is invalid for N > (cid:4) Remark 4.7
In fact, the proof of Proposition 4.6 shows that Conjecture 4.3 is satisfied wheneverthe inclusion S N < S + N is maximal. (cid:7) We are now ready for the main technical result of this section.
Proposition 4.8
Let M ≥ and N ≥ M . If S + M satisfies Conjecture 4.3 then so does S + N . Proof
Let A = A ( S + N ). We have to argue that under the hypothesis, the Hopf image A → A π of(11) is all of A . Since the quantum group attached to A π clearly contains S N , Proposition 3.10reduces the problem to showing that it also contains a quantum subgroup G < S + N that is ( M, N )-large in the sense of Definition 3.8.For this, fix a collection of rank-one projections P i , 0 ≤ i ≤ N − M N summing up to 1. Weform an N × N Latin square ( L ij ) ≤ i,j ≤ N − all of whose entries are the projections P i as follows: • if i, j ≤ M − L ij = P ( i − j ) mod M ; • we fill the rest of the first M rows with P i ’s arbitrarily so as to retain the Latin rectangleproperty (this is possible because 2 M ≤ N ); • complete the above Latin rectangle to a Latin square, once more using [1, Chapter 35, Lemma1].Having constructed L as above, consider the subspace Y ⊂ X N consisting of those N × N bis-tochastic matrices M of rank-one projections that are identical to L outside of the upper left hand M × M corner.Setting P = M − X i =0 P i , all operators appearing as entries of matrices M ∈ Y commute with P . Restricting these operatorsto the range of P (which is in turn isomorphic to C M ), we obtain the upper right hand arrow inthe composition A ( S + N ) M N ( C ( X N )) M N ( C ( Y )) M M ( C ( Y )) A ( S + M ) ∗ A ( S + N − M ) π η where the lower factorization occurs because by construction the off-block-diagonal entries L ij withprecisely one of i, j in { , · · · , M − } are projections orthogonal to P and hence vanish on Im P .21ur goal is now to show that the Hopf image of η in the above diagram contains an ( M, N )-largequantum subgroup G < S + M ∗ S + T < S + N , T := N − M. Equivalently, this means proving that the composition A ( S + M ) A ( S + M ) ∗ A ( S + T ) M M ( C ( Y )) η (13)is inner faithful. To verify this, recall that by construction the upper left hand M × M corners ofmatrices in Y are arbitrary bistochastic matrices in M M ∼ = End(Im P ) . Since the other entries of matrices in Y are identical to those of the fixed Latin square L , we have anisomorphism M M ( C ( Y )) ∼ = M M ( C ( X M )). The isomorphism described here renders (13) identicalto the canonical universal flat representation of A ( S + M ), which is inner faithful by hypothesis. (cid:4) As a consequence, we can prove Conjecture 4.3 for almost all N . Corollary 4.9
All S + N with N ≤ and N ≥ satisfy Conjecture 4.3. Proof
As explained above, we already know the conjecture to hold in the cases N ≤
4. For theremaining cases , it is enough to prove that S +5 satisfies Conjecture 4.3 in view of Proposition 4.8,and this is taken care of by Proposition 4.6. (cid:4) ∗ -algebras Corollary 4.9 shows that the universal flat matrix model is inner faithful for most quantum permu-tation groups. In this final section we show that we can do even better: it turns out that for thesame values of the parameter N a single finite-dimensional representation suffices to achieve innerfaithfulness. We first recall the relevant concept from [2, Definition 5.1]. Definition 4.10
A Hopf ∗ -algebra A is inner unitary if it has an inner faithful ∗ -homomorphisminto a finite-dimensional C ∗ -algebra. (cid:7) The main result of this subsection is the following improvement on Corollary 4.9.
Theorem 4.11
The Hopf ∗ -algebra A = A ( S + N ) is inner unitary for all N outside the range [6 , . Proof
We first tackle the smaller- N cases. (Case 1: N ≤ ) S + N is classical, and hence the conclusion follows from [9, Proposition 5.5]. (Case 2: N = 4 , ) Let x ∈ X N be any of the bistochastic matrices whose entries generate anon-commutative subalgebra of M N ( C ) and let y ∈ X classN ⊂ X N be such that the correspondingflat representation π classy is inner faithful on A ( S N ) (such x and y exist by Lemma 4.5) and [7,Proposition 5.3], respectively). The Hopf image of the representation π x ⊕ π y : A ( S + N ) → M N ( C ) ⊕ M N ( C )is then a non-commutative quotient Hopf ∗ -algebra of A containing S N . Since for N = 4 , S N < G < S + N , π x ⊕ π y is all of A , as desired. (Case 3: N ≥ ) The proof of Proposition 4.8 in fact shows that if 5 ≤ M ≤ N and A ( S + M )admits a finite inner faithful family { π z , . . . , π z n } ( z i ∈ X M ) of flat M -dimensional representations,then A admits a finite family { π z ′ , . . . , π z ′ n } ( z ′ i ∈ X N ) of flat representations whose joint Hopf imagesurjects onto A ( S + M ) ∗ A ( S + N − T ). Since for M = 5 we do have such a family { π x , π y } by the previousstep of the current proof, we have such an x ′ , y ′ ∈ X N .Further choosing any inner faithful flat representation π classw : A ( S N ) → M N ( C ) ( w ∈ X classN ),the resulting direct sum representation π x ′ ⊕ π y ′ ⊕ π w : A ( S + N ) → M N ( C ) ⊕ is inner faithful. Indeed, the quantum subgroup of S + N dual to its Hopf image contains both S N and S + M ∗ S + N − M and hence coincides with S + N by Corollary 3.4. (cid:4) References [1] Martin Aigner and G¨unter M. Ziegler.
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Michael Brannan, Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA
E-mail address : [email protected] Alexandru Chirvasitu, Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail address : [email protected] Amaury Freslon, Laboratoire de Math´ematiques dOrsay, Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay, 91405 Orsay, France
E-mail address : [email protected]@math.u-psud.fr