aa r X i v : . [ m a t h . QA ] A p r QUANTUM GROUPS, FROM A FUNCTIONAL ANALYSISPERSPECTIVE
TEODOR BANICA
Abstract.
It is well-known that any compact Lie group appears as closed subgroupof a unitary group, G ⊂ U N . The unitary group U N has a free analogue U + N , and thestudy of the closed quantum subgroups G ⊂ U + N is a problem of general interest. Wereview here the basic tools for dealing with such quantum groups, with all the neededpreliminaries included, and we discuss as well a number of more advanced topics. Dedicated to the memory of Stefan Banach.
Contents
Introduction 11. Operator algebras 32. Quantum groups 63. Representation theory 104. Basic examples 145. Reflection groups 176. Classification results 217. Maximal tori 248. Matrix models 27References 30
Introduction
The unitary group U N has a free analogue U + N , whose standard coordinate functions u ij ∈ C ( U + N ) still form a unitary matrix, whose transpose is unitary too, but no longercommute. To be more precise, consider the following universal C ∗ -algebra: C ( U + N ) = C ∗ (cid:16) ( u ij ) i,j =1 ,...,N (cid:12)(cid:12)(cid:12) u ∗ = u − , u t = ¯ u − (cid:17) Mathematics Subject Classification.
Key words and phrases.
Quantum group, Operator algebra.
This algebra has a comultiplication, a counit and an antipode, constructed by usingthe universal property of C ( U + N ), according to the following formulae:∆( u ij ) = X k u ik ⊗ u kj , ε ( u ij ) = δ ij , S ( u ij ) = u ∗ ji Thus U + N is a compact quantum group in the sense of Woronowicz [51], [52]. We areinterested here in the closed subgroups G ⊂ U + N , the main examples being:(1) The compact Lie groups, G ⊂ U N .(2) The duals G = b Γ of the finitely generated groups Γ = < g , . . . , g N > .(3) Deformations of the compact Lie groups, with parameter q = − G ⊂ U + N , whichsimplifies a number of things. There is as well some supplementary material, concerningsome additional methods, which are more recent and specialized. Disclaimer.
The closed subgroups G ⊂ U + N that we discuss here do not cover severalinteresting examples of “functional analytic” quantum groups, such as:(1) The q -deformations of the compact Lie groups G ⊂ U N , with general Drinfeldparameter q ∈ T , or with general Woronowicz parameter q ∈ R . For a functionalanalytic treatment of these quantum groups, we refer to [49], [51].(2) The locally compact groups G ⊂ GL N ( C ), and their various known quantum groupversions. For some general theory and some examples here, struggling howeverwith the existence of the Haar measure, we refer to [33], [40].(3) The quantum group type objects underlying the combinatorics of certain “speciallypatterned” random matrices. The subject here is very wide, and for a number ofsuch constructions, we refer for instance to [30], [37].We believe, however, that there is a way of linking the present material with all theseconstructions. Regarding (1), a natural idea here, which has not been tried yet, wouldbe that of studying first the q -deformations of the easy quantum groups, at q = ± i . Re-garding (2), the half-liberation machinery from [19] applies a priori to the locally compactcase as well, and this remains to be explored. Finally, regarding (3), some advances hereshould normally come along the lines of [12], and of subsequent papers.Regarding the possible applications of all this, the problem is open. Our belief is thatthe closed subgroups G ⊂ U + N , and the theory that has been developed for them, can beof help, in connection with a number of questions in quantum physics. Unfortunately, wehave nothing concrete for the moment. This is a matter of time. UANTUM GROUPS 3
The paper is organized in 4 parts, as follows:(1) Sections 1-2 are an introduction to the closed subgroups G ⊂ U + N , with the mainexamples ( O N , O ∗ N , O + N , U N , U ∗ N , U + N ) explained in detail.(2) Sections 3-4 are a presentation of Woronowicz’s theory in [50], [51], with the mainexamples and their twists ( ¯ O N , ¯ O ∗ N , O + N , ¯ U N , ¯ U ∗ N , U + N ) worked out.(3) Sections 5-6 contain more specialized results, regarding the corresponding reflec-tion groups ( H N , H ∗ N , H + N , K N , K ∗ N , K + N ), and other quantum groups.(4) Sections 7-8 contain results regarding the closed subgroups H ⊂ G , the maximaltori b Γ ⊂ G , and the matrix models π : C ( G ) → M K ( C ( X )). Acknowledgements.
I would like to thank Julien Bichon, Alex Chirvasitu, BenoˆıtCollins, Steve Curran, Uwe Franz, Ion Nechita, Adam Skalski, Roland Speicher andRoland Vergnioux, for substantial joint work on the subject, and for their heavy influenceon the point of view developed here. Thanks to Poufinette, too.1.
Operator algebras
In order to introduce the compact quantum groups, we will use the space/algebracorrespondence coming from operator algebra theory. Let us begin with:
Proposition 1.1.
Given a Hilbert space H , the linear operators T : H → H which arebounded, in the sense that || T || = sup || x ||≤ || T x || is finite, form a complex algebra withunit, denoted B ( H ) . This algebra has the following properties: (1) B ( H ) is complete with respect to || . || , and so we have a Banach algebra. (2) B ( H ) has an involution T → T ∗ , given by < T x, y > = < x, T ∗ y > .In addition, the norm and the involution are related by the formula || T T ∗ || = || T || .Proof. The fact that we have indeed an algebra follows from: || S + T || ≤ || S || + || T || , || λT || = | λ | · || T || , || ST || ≤ || S || · || T || Regarding now (1), if { T n } ⊂ B ( H ) is Cauchy then { T n x } is Cauchy for any x ∈ H , sowe can define the limit T = lim n →∞ T n by setting T x = lim n →∞ T n x .As for (2), here the existence of T ∗ comes from the fact that φ ( x ) = < T x, y > beinga linear map H → C , we must have φ ( x ) = < x, T ∗ y > , for a certain vector T ∗ y ∈ H .Moreover, since this vector is unique, T ∗ is unique too, and we have as well:( S + T ) ∗ = S ∗ + T ∗ , ( λT ) ∗ = ¯ λT ∗ , ( ST ) ∗ = T ∗ S ∗ , ( T ∗ ) ∗ = T Observe also that we have indeed T ∗ ∈ B ( H ), because: || T || = sup || x || =1 sup || y || =1 < T x, y > = sup || y || =1 sup || x || =1 < x, T ∗ y > = || T ∗ || T. BANICA
Regarding the last assertion, we have || T T ∗ || ≤ || T || · || T ∗ || = || T || . Also, we have: || T || = sup || x || =1 | < T x, T x > | = sup || x || =1 | < x, T ∗ T x > | ≤ || T ∗ T || By replacing T → T ∗ we obtain from this || T || ≤ || T T ∗ || , and we are done. (cid:3) In what follows we will be interested in the algebras of operators, rather than in theoperators themselves. The axioms here, coming from Proposition 1.1, are as follows:
Definition 1.2.
A unital C ∗ -algebra is a complex algebra with unit A , having a norm a → || a || which makes it into a Banach algebra (the Cauchy sequences converge), andhaving as well an involution a → a ∗ , which satisfies || aa ∗ || = || a || , for any a ∈ A . As a basic example, B ( H ) is a C ∗ -algebra. More generally, any closed ∗ -subalgebra A ⊂ B ( H ) is a C ∗ -algebra. The celebrated Gelfand-Naimark-Segal (GNS) theorem statesthat any C ∗ -algebra appears in fact in this way. We will be back to this later on.One very interesting feature of Definition 1.2, making the link with several branches ofabstract mathematics, comes from the following basic observation: Proposition 1.3. If X is an abstract compact space, the algebra C ( X ) of continuousfunctions f : X → C is a C ∗ -algebra, with norm || f || = sup x ∈ X | f ( x ) | , and involution f ∗ ( x ) = f ( x ) . This algebra is commutative: f g = gf , for any f, g ∈ C ( X ) .Proof. Almost everything here is trivial. Observe also that we have indeed: || f f ∗ || = sup x ∈ X | f ( x ) f ( x ) | = sup x ∈ X | f ( x ) | = || f || Finally, we have f g = gf , since f ( x ) g ( x ) = g ( x ) f ( x ) for any x ∈ X . (cid:3) In order to work out the precise space/algebra correspondence coming from Proposition1.3, we will need some basic spectral theory. Let us begin with:
Definition 1.4.
Given a complex algebra A , the spectrum of an element a ∈ A is σ ( a ) = (cid:8) λ ∈ C (cid:12)(cid:12) a − λ A − (cid:9) where A − ⊂ A is the set of invertible elements. As a basic example, the spectrum of a usual matrix M ∈ M N ( C ) is the collection of itseigenvalues. Also, the spectrum of a continuous function f ∈ C ( X ) is its image.Given an element a ∈ A , its spectral radius ρ ( a ) is by definition the radius of thesmallest disk centered at 0 containing σ ( a ). We have the following result: Proposition 1.5.
Let A be a C ∗ -algebra. (1) The spectrum of a norm one element is in the unit disk. (2)
The spectrum of a unitary element ( a ∗ = a − ) is on the unit circle. (3) The spectrum of a self-adjoint element ( a = a ∗ ) consists of real numbers. (4) The spectral radius of a normal element ( aa ∗ = a ∗ a ) is equal to its norm. UANTUM GROUPS 5
Proof.
Here (1) is clear, by using the formula 1 / (1 − x ) = 1 + x + x + . . . Regarding now the middle assertions, we can use here the elementary fact that if f isa rational function having poles outside σ ( a ), then σ ( f ( a )) = f ( σ ( a )). Indeed, by usingthe functions z − and ( z + it ) / ( z − it ), we obtain the results.From (1) we obtain ρ ( a ) ≤ || a || . For the converse, if we fix ρ > ρ ( a ), we have: Z | z | = ρ z n z − a dz = ∞ X k =0 (cid:18)Z | z | = ρ z n − k − dz (cid:19) a k = a n By applying the norm and taking n -th roots we get ρ ≥ lim || a n || /n .In the case a = a ∗ we have || a n || = || a || n for any exponent of the form n = 2 k , and bytaking n -th roots we get ρ ≥ || a || . This gives the missing inequality ρ ( a ) ≥ || a || .In the general case aa ∗ = a ∗ a we have a n ( a n ) ∗ = ( aa ∗ ) n , and we get ρ ( a ) = ρ ( aa ∗ ).Now since aa ∗ is self-adjoint, we get ρ ( aa ∗ ) = || a || , and we are done. (cid:3) We are now in position of proving a key result:
Theorem 1.6 (Gelfand) . Any commutative C ∗ -algebra is the form C ( X ) , with its “spec-trum” X = Spec ( A ) appearing as the space of characters χ : A → C .Proof. Given a commutative C ∗ -algebra A , we can define indeed X to be the set ofcharacters χ : A → C , with the topology making continuous all the evaluation maps ev a : χ → χ ( a ). Then X is a compact space, and a → ev a is a morphism of algebras ev : A → C ( X ). We first prove that ev is involutive. We use the following formula: a = a + a ∗ − i · i ( a − a ∗ )2Thus it is enough to prove the equality ev a ∗ = ev ∗ a for self-adjoint elements a . But thisis the same as proving that a = a ∗ implies that ev a is a real function, which is in turntrue, because ev a ( χ ) = χ ( a ) is an element of σ ( a ), contained in R .Since A is commutative, each element is normal, so ev is isometric, || ev a || = ρ ( a ) = || a || .It remains to prove that ev is surjective. But this follows from the Stone-Weierstrasstheorem, because ev ( A ) is a closed subalgebra of C ( X ), which separates the points. (cid:3) In view of Gelfand’s theorem, we can now formulate:
Definition 1.7.
Given an arbitrary C ∗ -algebra A , we write A = C ( X ) , and call X a non-commutative compact space. Equivalently, the category of the noncommutative compactspaces is the category of the C ∗ -algebras, with the arrows reversed. When A is commutative, the space X considered above exists indeed, as a Gelfandspectrum, X = Spec ( A ). In general, X is something rather abstract, and our philosophyhere will be that of studying of course A , but formulating our results in terms of X . Forinstance whenever we have a morphism Φ : A → B , we will write A = C ( X ) , B = C ( Y ),and rather speak of the corresponding morphism φ : Y → X . And so on. T. BANICA
Finally, let us review the other fundamental result regarding the C ∗ -algebras, namelythe representation theorem of Gelfand, Naimark and Segal. We will need: Proposition 1.8.
For an element a ∈ A , the following are equivalent: (1) a is positive, in the sense that σ ( a ) ⊂ [0 , ∞ ) . (2) a = b , for some b ∈ A satisfying b = b ∗ . (3) a = cc ∗ , for some c ∈ A .Proof. Regarding (1) = ⇒ (2), observe that σ ( a ) ⊂ R implies a = a ∗ . Thus the algebra < a > is commutative, and by using the Gelfand theorem, we can set b = √ a .The implication (2) = ⇒ (3) is trivial, because we can set c = b . Observe that(2) = ⇒ (1) is clear too, because we have σ ( a ) = σ ( b ) = σ ( b ) ⊂ R = [0 , ∞ ).For (3) = ⇒ (1), we proceed by contradition. By multiplying c by a suitable element of < cc ∗ > , we are led to the existence of an element d = 0 satisfying − dd ∗ ≥
0. By writing d = x + iy with x = x ∗ , y = y ∗ we have dd ∗ + d ∗ d = 2( x + y ) ≥
0, and so d ∗ d ≥
0. Butthis contradicts the elementary fact that σ ( de ) , σ ( ed ) must coincide outside { } . (cid:3) We will need as well the following definition:
Definition 1.9.
Consider the linear continuous maps ϕ : A → C , called states of A . (1) ϕ is called positive when a ≥ ⇒ ϕ ( a ) ≥ . (2) ϕ is called faithful and positive when a ≥ , a = 0 = ⇒ ϕ ( a ) > . In the commutative case, A = C ( X ), the states are of the form ϕ ( f ) = R X f ( x ) dµ ( x ),with µ being positive/strictly positive in order for ϕ to be positive/faithful and positive.In analogy with the fact that any compact space X has a probability measure µ , one canprove that any C ∗ -algebra A has a faithful positive state ϕ : A → C . See [39].With these ingredients in hand, we can now state: Theorem 1.10 (GNS theorem) . Let A be a C ∗ -algebra. (1) A appears as a closed ∗ -subalgebra A ⊂ B ( H ) , for some Hilbert space H . (2) When A is separable (usually the case), H can be chosen to be separable. (3) When A is finite dimensional, H can be chosen to be finite dimensional.Proof. In the commutative case, where A = C ( X ), this algebra can be represented on H = L ( X ), via T f ( g ) = f g , provided that we have a probability measure on X .In general now, we can pick a faithful positive state ϕ : A → C , then let H = l ( A )be the completion of A with respect to the scalar product < a, b > = ϕ ( ab ∗ ), and finallyrepresent A on this space via T a ( b ) = ab . For details here, we refer to [39]. (cid:3) Quantum groups
The quantum groups are abstract objects, generalizing the usual groups. The mostbasic examples are the group duals. Let us recall indeed that associated to any discretegroup Γ is its group algebra C ∗ (Γ), obtained as enveloping C ∗ -algebra of the usual group UANTUM GROUPS 7 ∗ -algebra C [Γ] = span (Γ). This algebra has a canonical trace, given by tr ( g ) = δ g, onthe generators. With these conventions, we have the following result: Proposition 2.1.
Let G be a compact abelian group, and Γ = b G be its Pontrjagin dual,formed by the characters χ : G → T . We have then a Fourier transform isomorphism C ( G ) ≃ C ∗ (Γ) which transforms the comultiplication, counit and antipode of C ( G ) , given by ∆ ϕ ( g, h ) = ϕ ( gh ) , ε ( ϕ ) = ϕ (1) , Sϕ ( g ) = ϕ ( g − ) into the comultiplication, counit and antipode of C ∗ (Γ) , given on generators by: ∆( g ) = g ⊗ g , ε ( g ) = 1 , S ( g ) = g − Moreover, the Haar integration over G corresponds to the canonical trace of C ∗ (Γ) .Proof. The first assertion follows from the basic properties of the Pontrjagin duality, andfrom the Gelfand theorem. The proof of the second assertion, regarding ∆ , ε, S , is routine.As for the last assertion, regarding the standard traces, this is clear too. (cid:3)
In the non-abelian case now, there is still of lot of similarity between the algebras oftype C ( G ), and those of type C ∗ (Γ). As a basic result here, we have: Proposition 2.2.
The comultiplication, counit and antipode of both the algebras C ( G ) ,with G compact group, and C ∗ (Γ) , with Γ discrete group, satisfy: (1) (∆ ⊗ id )∆ = ( id ⊗ ∆)∆ . (2) ( id ⊗ ε )∆ = ( ε ⊗ id )∆ = id . (3) m ( id ⊗ S )∆ = m ( S ⊗ id )∆ = ε ( . )1 In addition, in both cases the square of the antipode is the identity, S = id .Proof. For the algebras of type C ( G ), all the above formulae are well-known, coming viathe Gelfand space/algebra correspondence from the following identities:( gh ) k = g ( hk ) , g g = g , gg − = g − g = 1 , ( g − ) − = g As for the algebras of type C ∗ (Γ), the formulae in the statement are all trivial. (cid:3) The above results suggest looking for a joint axiomatization of the algebras of type C ( G ) and C ∗ (Γ). However, this is quite tricky, due to some analytic issues with (3).Instead of getting into this, we will assume that G is a compact Lie group, and that Γ isa finitely generated group. These assumptions are related, due to: Proposition 2.3.
Given a compact abelian group G , and a discrete abelian group Γ ,related by Pontrjagin duality, G = b Γ and Γ = b G , the following are equivalent: (1) G is a compact Lie group, G ⊂ U N . (2) Γ is finitely generated, Γ = < g , . . . , g N > . T. BANICA
Proof.
Assuming that we have a representation π : G ⊂ U N , we know from Peter-Weyltheory that π decomposes as a sum of characters, π = g ⊕ . . . ⊕ g N , and we obtain inthis way generators for Γ = b G . Conversely, assuming Γ = < g , . . . , g N > , the direct sum π = g ⊕ . . . ⊕ g N is a faithful representation π : G ⊂ U N of the dual G = b Γ. (cid:3) With these observations in hand, we can now go ahead, and formulate:
Definition 2.4.
A Woronowicz algebra is a C ∗ -algebra A , given with a unitary matrix u ∈ M N ( A ) whose coefficients generate A , such that: (1) ∆( u ij ) = P k u ik ⊗ u kj defines a morphism of C ∗ -algebras A → A ⊗ A . (2) ε ( u ij ) = δ ij defines a morphism of C ∗ -algebras A → C . (3) S ( u ij ) = u ∗ ji defines a morphism of C ∗ -algebras A → A opp .In this case, we write A = C ( G ) = C ∗ (Γ) and call G a compact matrix quantum group,and Γ a finitely generated discrete quantum group. Also, we write G = b Γ , Γ = b G . As a basic example, we have the algebra A = C ( G ), with G ⊂ U N , with the matrixformed by the standard coordinates u ij ( g ) = g ij . Indeed, ∆ , ε, S are given by the formulaein the statement, due to the following formulae, valid for the unitary matrices:( U V ) ij = X k U ik V kj , ij = δ ij , ( U − ) ij = ¯ U ji The other basic example is the algebra A = C ∗ (Γ), with Γ = < g , . . . , g N > , togetherwith the matrix u = diag ( g , . . . , g N ). Indeed, the comultiplication, counit and antipodeof this algebra are by definition given by the formulae in the statement.As a first general result now, regarding the above objects, we have: Proposition 2.5.
For a Woronowicz algebra, the maps ∆ , ε, S satisfy the usual conditionsfor a comultiplication, counit and antipode, as in Proposition 2.2 above, at least on thedense ∗ -subagebra A ⊂ A generated by the coordinates u ij .Proof. This is clear, because all the three formulae in Proposition 2.2, as well as thesupplementary formula S = id , are trivially satisfied on the generators u ij . By linearityand multiplicativity, all these formulae are therefore satisfied on A . (cid:3) At the level of the new examples now, we first have:
Proposition 2.6.
The following universal algebras are Woronowicz algebras, C ( O + N ) = C ∗ (cid:16) ( u ij ) i,j =1 ,...,N (cid:12)(cid:12)(cid:12) u = ¯ u, u t = u − (cid:17) C ( U + N ) = C ∗ (cid:16) ( u ij ) i,j =1 ,...,N (cid:12)(cid:12)(cid:12) u ∗ = u − , u t = ¯ u − (cid:17) and so the underlying noncommutative spaces O + N , U + N are compact quantum groups. UANTUM GROUPS 9
Proof.
This follows from the fact that if a matrix u is orthogonal/biunitary, then so arethe matrices u ∆ ij = P k u ik ⊗ u kj , u εij = δ ij , u Sij = u ∗ ji . Thus, we can define maps ∆ , ε, S asin Definition 2.4, by using the universal properties of C ( O + N ), C ( U + N ). See [46]. (cid:3) The basic properties of O + N , U + N can be summarized as follows: Proposition 2.7.
The quantum groups O + N , U + N have the following properties: (1) The closed subgroups G ⊂ U + N are exactly the N × N compact quantum groups. Asfor the closed subgroups G ⊂ O + N , these are those satisfying u = ¯ u . (2) We have “liberation” embeddings O N ⊂ O + N and U N ⊂ U + N , obtained by dividingthe algebras C ( O + N ) , C ( U + N ) by their respective commutator ideals. (3) We have as well embeddings b L N ⊂ O + N and b F N ⊂ U + N , where L N is the free productof N copies of Z , and where F N is the free group on N generators.Proof. Here (1) is clear from definitions, with the remark that, in the context of Definition2.4 above, the formula S ( u ij ) = u ∗ ji shows that ¯ u must be unitary too. The assertion (2)follows from the Gelfand theorem. Finally, (3) follows from definitions, with the remarkthat in a group algebra we have ¯ g = g − , and so g = ¯ g if and only if g = 1. (cid:3) In order to construct more examples, we can look for intermediate objects for theinclusion U N ⊂ U + N . There are several possible choices here, and the “simplest” ones,from a point of view that will be explained in detail later on, are as follows: Proposition 2.8.
We have intermediate quantum groups as follows: (1) O N ⊂ O ∗ N ⊂ O + N , obtained from O + N by imposing to the variables u ij the half-commutation relations abc = cba . (2) U N ⊂ U ∗ N ⊂ U + N , obtained from U + N by imposing to the variables u ij , u ∗ ij the half-commutation relations abc = cba .Proof. This is elementary, by using the fact that if the entries of u = ( u ij ) half-commute,then so do the entries of u ∆ ij = P k u ik ⊗ u kj , u εij = δ ij , u Sij = u ∗ ji . See [16], [19]. (cid:3) In order to distinguish the various quantum groups that we have, we can use:
Proposition 2.9.
Given a closed subgroup G ⊂ U + N , consider its “diagonal torus”, whichis the closed subgroup T ⊂ G constructed as follows: C ( T ) = C ( G ) . D u ij = 0 (cid:12)(cid:12)(cid:12) ∀ i = j E This torus is then a group dual, T = b Λ , where Λ = < g , . . . , g N > is the discrete groupgenerated by the elements g i = u ii , which are unitaries inside C ( T ) .Proof. Since u is unitary, its diagonal entries g i = u ii are unitaries inside C ( T ). Moreover,from ∆( u ij ) = P k u ik ⊗ u kj we obtain ∆( g i ) = g i ⊗ g i , and so these unitaries g i ∈ C ( T )are group-like. We conclude that we have C ( T ) = C ∗ (Λ), and so T = b Λ. (cid:3) Now back to our basic examples of quantum groups, we can formulate:
Theorem 2.10.
The basic examples of compact quantum groups are as follows, U N / / U ∗ N / / U + N O N / / O O O ∗ N / / O O O + N O O and these quantum groups are non-isomorphic, distinguished by their diagonal tori.Proof. The fact that we have quantum groups as above is something that we already know.Regarding now the diagonal tori, these are as follows, with ◦ being by the half-classicalproduct, subject to relations abc = cba between the standard generators: c Z N / / d Z ◦ N / / d Z ∗ N c Z N / / O O d Z ◦ N / / O O d Z ∗ N O O Now since the discrete groups in this diagram are clearly non-isomorphic, this showsthat the corresponding quantum groups are non-isomorphic as well, as claimed. (cid:3) Representation theory
In order to reach to some more advanced insight into the structure of the closed sub-groups G ⊂ U + N , we can use representation theory. Let us begin with: Definition 3.1.
Let ( A, u ) be a Woronowicz algebra, and consider its dense ∗ -subalgebra A ⊂ A of “smooth elements”, generated by the standard coordinates u ij . (1) A corepresentation of A is a unitary matrix r ∈ M n ( A ) satisfying ∆( r ij ) = P k r ik ⊗ r kj , ε ( r ij ) = δ ij and S ( r ij ) = r ∗ ji . (2) The corepresentations are subject to making sums, r + p = diag ( r, p ) , tensor prod-ucts, ( r ⊗ p ) ia,jb = r ij p ab , and taking conjugates, (¯ r ) ij = r ∗ ij . (3) Given r ∈ M n ( A ) , p ∈ M m ( A ) we set Hom ( r, p ) = { T ∈ M m × n ( C ) | T r = pT } , andwe use the notations F ix ( r ) = Hom (1 , r ) , and End ( r ) = Hom ( r, r ) . (4) Two corepresentations r ∈ M n ( A ) , p ∈ M m ( A ) are called equivalent, and we write r ∼ p , when n = m , and Hom ( r, p ) contains an invertible element. For A = C ( G ) we obtain in this way the representations of G , as a consequence of theGelfand space/algebra correspondence. For A = C ∗ (Γ), observe that any group element UANTUM GROUPS 11 g ∈ Γ is a one-dimensional corepresentation. We will see later on that, up to equivalence,each corepresentation of C ∗ (Γ) splits as a direct sum of group elements.We will need as well the following standard fact: Proposition 3.2.
The characters of corepresentations, given by χ r = P i r ii , satisfy: χ r + p = χ r + χ p , χ r ⊗ p = χ r χ p , χ ¯ r = χ ∗ r In addition, given two equivalent corepresentations, r ∼ p , we have χ r = χ p .Proof. The three formulae in the statement are all clear from definitions. Regarding nowthe last assertion, assuming that we have r = T − pT , we obtain: χ r = T r ( r ) = T r ( T − pT ) = T r ( p ) = χ p We conclude that r ∼ p implies χ r = χ p , as claimed. (cid:3) In order to work out the analogue of the Peter-Weyl theory, we need to integrate over G . Things here are quite tricky, and best is to start with a definition, as follows: Definition 3.3.
The Haar integration of a Woronowicz algebra A = C ( G ) is the uniquepositive unital tracial state R G : A → C subject to the invariance conditions (cid:18)Z G ⊗ id (cid:19) ∆ = (cid:18) id ⊗ Z G (cid:19) ∆ = Z G ( . )1 provided that such a state exists indeed, and is unique. As a basic example, given a compact Lie group G ⊂ U N , the algebra A = C ( G ) hasindeed a Haar integration, which is the integration with respect to the Haar measure of G .Moreover, this latter measure can be obtained by starting with any probability measureon G , and then performing a Ces`aro limit with respect to the convolution.In analogy with this fact, we have the following result: Theorem 3.4.
Any Woronowicz algebra has a Haar integration, which can be constructedby starting with any faithful positive unital state ϕ ∈ A ∗ , and taking the Ces`aro limit Z G = lim n →∞ n n X k =1 ϕ ∗ k where the convolution operation for states is given by φ ∗ ψ = ( φ ⊗ ψ )∆ .Proof. As already mentioned, this is well-known in the commutative case, A = C ( G ).In the group dual case, A = C ∗ (Γ), the result follows from Proposition 2.1 when Γ isabelian, the Haar functional being given by R b Γ g = δ g, , for any g ∈ Γ. When Γ is nolonger abelian, the result still holds, and this is something well-known, and standard.In the general case now, everything is quite tricky. We refer here to Woronowicz’spaper [50], and also to the paper of Maes and Van Daele [35], with the remark that ourassumption G ⊂ U + N , which implies S = id , simplifies quite a number of things. (cid:3) Now back to representations, the basic integration result that we will need is:
Proposition 3.5.
For any corepresentation r ∈ M n ( A ) , the operator P = (cid:18) id ⊗ Z G (cid:19) r ∈ M n ( C ) is the orthogonal projection onto the space F ix ( r ) = { x ∈ C n | r ( x ) = x ⊗ } .Proof. The invariance conditions in Definition 3.3 applied to ϕ = r ij read: X k r ik P kj = X k P ik r kj = P ij Thus we have rP = P r = P , and this gives the result. See [50]. (cid:3) With these results in hand, we can now develop the Peter-Weyl theory:
Theorem 3.6.
The corepresentations of a Woronowicz algebra ( A, u ) , taken moduloequivalence, are subject to the following Peter-Weyl type results: (1) Each representation appears as a sum of irreducible corepresentations. Also, eachirreducible corepresentation appears inside a tensor product of u, ¯ u . (2) The characters of irreducible corepresentations have norm , and are pairwiseorthogonal with respect to the scalar product < a, b > = R G ab ∗ . (3) The dense subalgebra
A ⊂ A decomposes as a direct sum A = ⊕ r ∈ Irr ( A ) M dim( r ) ( C ) ,with the summands being pairwise orthogonal with respect to <, > .Proof. This follows as in the classical case, with the various statements involving the Haarfunctional basically coming from Proposition 3.5 above. See Woronowicz [51]. (cid:3)
As a first application of these results, we have:
Theorem 3.7.
Let A full be the enveloping C ∗ -algebra of A , and let A red be the quotientof A by the null ideal of the Haar integration. The following are then equivalent: (1) The Haar functional of A full is faithful. (2) The projection map A full → A red is an isomorphism. (3) The counit map ε : A full → C factorizes through A red . (4) Kesten criterion: N ∈ σ ( Re ( χ u )) , inside the algebra A red .If this is the case, we say that the underlying discrete quantum group Γ is amenable.Proof. This is well-known in the group dual case, A = C ∗ (Γ), with Γ being a usual discretegroup. In general, the result follows by adapting the group dual case proof, by replacingwhere needed group elements by irreducible corepresentations. See [38]. (cid:3) The above results suggest that the “combinatorics” of a discrete quantum group Γ,appearing via A = C ∗ (Γ), should come from the fusion rules on Irr ( A ). This is indeedthe case, and a whole theory can be developed here. See [16], [25], [29]. UANTUM GROUPS 13
Let us explain now Woronowicz’s Tannakian duality result from [52], in its “soft” form,worked out in [36]. The precise definition that we will need is:
Definition 3.8.
The Tannakian category associated to a Woronowicz algebra ( A, u ) isthe collection C = ( C kl ) of vector spaces C kl = Hom ( u ⊗ k , u ⊗ l ) where the tensor powers, taken with respect to colored integers k, l = ◦ • • ◦ . . . are givenby u ∅ = 1 , u ◦ = u , u • = ¯ u and multiplicativity. Observe that C is indeed a tensor category, and more precisely is a tensor subcategoryof the tensor category formed by the spaces M kl = L ( H ⊗ k , H ⊗ l ), where H ≃ C N is theHilbert space where u ∈ M N ( A ) coacts, and where the tensor powers H ⊗ k with k coloredinteger are defined by H ∅ = C , H ◦ = H, H • = ¯ H ≃ H and multiplicativity.The Tannakian duality result, in its “soft” form, is as follows: Theorem 3.9.
Given a Woronowicz algebra ( A, u ) with u ∈ M N ( A ) , with associatedTannakian category C = ( C kl ) , we have A = C ( U + N ) (cid:14) D T ∈ Hom ( v ⊗ k , v ⊗ l ) (cid:12)(cid:12)(cid:12) ∀ k, l, ∀ T ∈ C kl E where v denotes the fundamental corepresentation of C ( U + N ) .Proof. If we denote by A ′ the universal algebra on the right, we have a morphism A ′ → A ,because the canonical morphism C ( U + N ) → A factorizes through the ideal defining A ′ , bydefinition of the Tannakian category C = ( C kl ). Conversely now, the fact that we havean arrow A → A ′ follows from Woronowicz’s Tannakian duality results in [52], but thereis as well a direct, Hopf algebra proof for this, worked out in [36]. (cid:3) Generally speaking, knowing the Tannakian category of a Woronowicz algebra A solvesmost of the fundamental problems regarding A . As an illustration here, let us discuss thecomputation of the Haar state of A . The formula is very simple, as follows: Theorem 3.10.
Assuming that A = C ( G ) has Tannakian category C = ( C kl ) , the Haarintegration over G is given by the Weingarten type formula Z G u e i j . . . u e k i k j k = X π,σ ∈ D k δ π ( i ) δ σ ( j ) W k ( π, σ ) for any colored integer k = e . . . e k and any multi-indices i, j , where D k is a linear basisof C kk , δ π ( i ) = < π, e i ⊗ . . . ⊗ e i k > , and W k = G − k , with G k ( π, σ ) = < π, σ > .Proof. We know from Proposition 3.5 above that the integrals in the statement formaltogether the orthogonal projection P e onto the space F ix ( u ⊗ k ) = span ( D k ).By a standard linear algebra computation, it follows that we have P = W E , where E ( x ) = P π ∈ D k < x, ξ π > ξ π , and where W is the inverse on span ( T π | π ∈ D k ) of the restriction of E . But this restriction is the linear map given by G k , and so W is the linearmap given by W k , and this gives the formula in the statement. See [8]. (cid:3) Basic examples
We will show now that the Tannakian categories of the main 6 quantum groups appearin the simplest possible way: from certain “categories” of set-theoretic partitions.In order to explain this material, let us begin with:
Definition 4.1.
Let P ( k, l ) be the set of partitions between an upper colored integer k ,and a lower colored integer l . A set D = F k,l D ( k, l ) with D ( k, l ) ⊂ P ( k, l ) is called acategory of partitions when it is stable under the following operations: (1) The horizontal concatenation operation, ( π, σ ) → [ πσ ] . (2) The vertical concatenation ( π, σ ) → [ σπ ] , when the middle symbols match. (3) The upside-down turning operation ∗ , with switching of the colors, ◦ ↔ • . Here, in the definition of the second operation, we agree that the connected componentsthat can appear in the middle, when concatenating, are erased afterwards.The relation with the Tannakian categories comes from:
Proposition 4.2.
Each π ∈ P ( k, l ) produces a linear map T π : ( C N ) ⊗ k → ( C N ) ⊗ l , T π ( e i ⊗ . . . ⊗ e i k ) = X j ...j l δ π (cid:18) i . . . i k j . . . j l (cid:19) e j ⊗ . . . ⊗ e j l with the Kronecker type symbols δ π ∈ { , } depending on whether the indices fit or not.The assignement π → T π is categorical, in the sense that we have T π ⊗ T σ = T [ πσ ] , T π T σ = N c ( π,σ ) T [ σπ ] , T ∗ π = T π ∗ where c ( π, σ ) are certain integers, coming from the erased components in the middle.Proof. All three formulae are indeed elementary to establish. See [15]. (cid:3)
In relation now with the quantum groups, we have the following notion:
Definition 4.3.
A closed subgroup G ⊂ U + N is called easy when we have Hom ( u ⊗ k , u ⊗ l ) = span (cid:16) T π (cid:12)(cid:12)(cid:12) π ∈ D ( k, l ) (cid:17) for any colored integers k, l , for a certain category of partitions D ⊂ P . As we will see in what follows, this formalism covers many interesting examples ofgroups and quantum groups. Let us first go back to the examples that we have. Thesequantum groups are all easy, coming from certain categories of pairings, as follows:
UANTUM GROUPS 15
Theorem 4.4.
The basic unitary quantum groups are all easy, as follows, U N / / U ∗ N / / U + N O N / / O O O ∗ N / / O O O + N O O : P (cid:15) (cid:15) P ∗ o o (cid:15) (cid:15) N C o o (cid:15) (cid:15) P P ∗ o o N C o o with the corresponding categories of partitions being the following ones: (1) P , N C are respectively the categories of pairings, and of noncrossing pairings,and P ∗ is the category of pairings having the property that, when the legs arerelabelled clockwise ◦ • ◦ • . . . , each string connects ◦ − • . (2) P is the category of pairings which are “matching”, in the sense that the verticalstrings connect either ◦ − ◦ or • − • , and the horizontal strings connect ◦ − • , and P ∗ = P ∩ P ∗ and N C = P ∩ N C .Proof. The results for O N , U N go back to Brauer’s paper [21], their free versions are workedout in [8], and the half-liberated results are from [16], [19], the idea being as follows:(1) U + N is defined via the relations u ∗ = u − , u t = ¯ u − , which tell us that the operators T π , with π = ∩◦• and π = ∩•◦ , must be in the associated Tannakian category C . Wetherefore obtain C = span ( T π | π ∈ D ), with D = < ∩◦• , ∩•◦ > = N C , as claimed.(2) O + N ⊂ U + N is defined by imposing the relations u ij = ¯ u ij , which tell us that theoperators T π , with π = | ◦• and π = | •◦ , must be in the associated Tannakian category C . Wetherefore obtain C = span ( T π | π ∈ D ), with D = < N C , | ◦• , | •◦ > = N C , as claimed.(3) U N ⊂ U + N is defined via the relations [ u ij , u kl ] = 0 and [ u ij , ¯ u kl ] = 0, which tellus that the operators T π , with π = / \ ◦◦◦◦ and π = / \ ◦••◦ , must be in the associated Tannakiancategory C . Thus C = span ( T π | π ∈ D ), with D = < N C , / \ ◦◦◦◦ , / \ ◦••◦ > = P , as claimed.(4) Regarding now U ∗ N ⊂ U + N , the corresponding Tannakian category is generated bythe operators T π , with π = / \| , taken with all the possible 2 = 8 matching colorings. Sincethese latter 8 partitions generate the category P ∗ , we obtain the result.(5) In order to deal now with O N , we can simply use the formula O N = O + N ∩ U N .At the categorical level, this tells us that the associated Tannakian category is given by C = span ( T π | π ∈ D ), with D = < N C , P > = P , as claimed.(6) Finally, for O ∗ N we can proceed similarly, by using the formula O ∗ N = O + N ∩ U ∗ N .At the categorical level, this tells us that the associated Tannakian category is given by C = span ( T π | π ∈ D ), with D = < N C , P ∗ > = P ∗ , as claimed. (cid:3) Let us discuss now some more examples, which are of the same nature as the 6 basicones. The idea is that we can twist the basic quantum groups, as follows:
Proposition 4.5.
We have quantum groups as follows, obtained via the twisted commu-tation relations ab = ± ba , and twisted half-commutation relations abc = ± cba , ¯ U N / / ¯ U ∗ N / / U + N ¯ O N / / O O ¯ O ∗ N / / O O O + N O O where the signs for ¯ U N correspond to anticommutation for distinct entries on rows andcolumns, and commutation otherwise, and the other signs come from functoriality.Proof. This is clear indeed, by proceeding as in the proof of Proposition 2.6, with thesigns for ¯ U ∗ N being those producing an inclusion ¯ U N ⊂ ¯ U ∗ N , and with those for ¯ O N , ¯ O ∗ N producing inclusions ¯ O N ⊂ ¯ U N and ¯ O ∗ N ⊂ ¯ U ∗ N . For details here, we refer to [1]. (cid:3) In order to study the easiness properties of these quantum groups, we will need:
Proposition 4.6.
We have a signature map ε : P even → {− , } , given by ε ( π ) = ( − c ,where c is the number of switches needed to make π noncrossing. In addition: (1) For π ∈ P erm ( k, k ) ≃ S k , this is the usual signature. (2) For π ∈ P we have ( − c , where c is the number of crossings. (3) For π ∈ P obtained from σ ∈ N C even by merging blocks, the signature is .Proof. The fact that the number c in the statement is well-defined modulo 2 is standard,and we refer here to [1]. As for the remaining assertions, these are as well from [1]:(1) For π ∈ P erm ( k, k ) the standard form is π ′ = id , and the passage π → id comes bycomposing with a number of transpositions, which gives the signature.(2) For a general π ∈ P , the standard form is of type π ′ = | . . . | ∪ ... ∪∩ ... ∩ , and the passage π → π ′ requires c mod 2 switches, where c is the number of crossings.(3) For a partition π ∈ P even coming from σ ∈ N C even by merging a certain number n of blocks, the fact that the signature is 1 follows by recurrence on n . (cid:3) We can make act the partitions in P even on tensors in a twisted way, as follows: Proposition 4.7.
Associated to any partition π ∈ P even ( k, l ) is the linear map ¯ T π ( e i ⊗ . . . ⊗ e i k ) = X σ ≤ π ε ( σ ) X j :ker( ij )= σ e j ⊗ . . . ⊗ e j l and the assignement π → ¯ T π is categorical.Proof. This is routine, by using ingredients from Proposition 4.6. See [1]. (cid:3)
With these notions in hand, we can now investigate the twists, as follows:
UANTUM GROUPS 17
Theorem 4.8.
The quantum groups from Proposition 4.5 appear as Schur-Weyl twists ofthe quantum groups in Theorem 4.4, in the sense that for ¯ G we have Hom ( u ⊗ k , u ⊗ l ) = span (cid:16) ¯ T π (cid:12)(cid:12)(cid:12) π ∈ D ( k, l ) (cid:17) for any colored integers k, l , where D ⊂ P ⊂ P even is the category of partitions for G . Inaddition, the diagonal tori for G, ¯ G coincide.Proof. All this is quite routine, by following the proof of Theorem 4.4 above, and addingsigns where needed. The final assertion is clear as well. For details, see [1]. (cid:3)
As a first application of all this, we have:
Theorem 4.9.
In the N → ∞ limit, the law of the main character χ u is as follows: (1) For O N , U N we obtain Gaussian/complex Gaussian variables. (2) For O ∗ N , U ∗ N we obtain “squeezed” versions of these variables. (3) For O + N , U + N we obtain semicircular/circular variables.In addition, the asymptotic law of χ u is invariant under Schur-Weyl twisting.Proof. We know from the Peter-Weyl theory that the moments of χ u are the dimensionsof the spaces F ix ( u ⊗ k ). Now since with N → ∞ the linear maps T π become linearlyindependent, the asymptotic moments of χ u count the corresponding pairings, and modulosome standard facts from classical and free probability, this gives the result. See [1]. (cid:3) At a more advanced level, the Weingarten integration formula from Theorem 3.10 takesa particularly simple form, the Gram matrix being given by G kN ( π, σ ) = N | π ∨ σ | , where | . | is the number of blocks. For applications of this formula, see [1], [8], [9], [15].5. Reflection groups
The quantum groups that we considered so far, namely O N , U N and their liberations andtwists, are obviously of “continuous” nature. In order to have as well “discrete” examples,the idea will be that of looking at the corresponding quantum reflection groups.Let us begin with a study of the quantum permutations. For this purpose, we will needthe following functional analytic description of the usual symmetric group: Proposition 5.1.
Consider the symmetric group S N . (1) The standard coordinates v ij ∈ C ( S N ) , coming from the embedding S N ⊂ O N givenby the permutation matrices, are given by v ij = χ ( σ | σ ( j ) = i ) . (2) The matrix v = ( v ij ) is magic, in the sense that its entries are orthogonal projec-tions, summing up to on each row and each column. (3) The algebra C ( S N ) is isomorphic to the universal commutative C ∗ -algebra gener-ated by the entries of a N × N magic matrix. Proof.
The assertions (1,2) are both clear. If we set A = C ∗ comm (( w ij ) i,j =1 ,...,N | w = magic),we have a quotient map A → C ( S N ), given by w ij → v ij . On the other hand, by usingthe Gelfand theorem we can write A = C ( X ), with X being a compact space, and byusing the coordinates w ij we have X ⊂ O N , and then X ⊂ S N . Thus we have as well aquotient map C ( S N ) → A given by v ij → w ij , and this gives (3). See Wang [47]. (cid:3) With the above result in hand, we can now formulate:
Proposition 5.2.
The following construction produces a Woronowicz algebra, C ( S + N ) = C ∗ (cid:16) ( u ij ) i,j =1 ,...,N (cid:12)(cid:12)(cid:12) u = magic (cid:17) and the corresponding closed subgroup S + N ⊂ O + N has the following properties: (1) S + N is the universal compact quantum group acting on { , . . . , N } . (2) We have an embedding S N ⊂ S + N , given by u ij → χ ( σ | σ ( j ) = i ) . (3) This embedding is an isomorphism at N = 1 , , , but not at N ≥ .Proof. Here the first assertion is standard, by using the elementary fact that if u = ( u ij )is magic, then so are the matrices u ∆ ij = P k u ik ⊗ u kj , u εij = δ ij , u Sij = u ∗ ji .Regarding (1), given a closed subgroup G ⊂ U + N , it is straightforward to check thatΦ( δ i ) = P j u ij ⊗ δ j defines a coaction map precisely when u = ( u ij ) is a magic corepre-sentation of C ( G ), and this gives the result. Also, (2) is clear from Proposition 5.1.Regarding now (3), it is elementary to show that the entries of a N × N magic matrix,with N ≤
3, must pairwise commute. At N = 4 now, consider the following matrix: U = p − p − p p q − q − q q This matrix is magic for any two projections p, q , and if we choose these projections asfor < p, q > to be not commutative, and infinite dimensional, we conclude that C ( S +4 ) isnot commutative and infinite dimensional as well, and so not isomorphic to C ( S ).Finally, at N ≥ S +4 ⊂ S + N , obtained at the levelof the corresponding magic matrices by u → diag ( u, N − ). See [46]. (cid:3) At the representation theory level, we have the following result:
Theorem 5.3.
The quantum groups S N , S + N have the following properties: (1) They are both easy, the corresponding categories being the category of all partitions P , and the category of all noncrossing partitions N C . (2) The corresponding asymptotic laws of the main characters are respectively thePoisson law, and the Marchenko-Pastur (or free Poisson) law.Proof.
The assertions here are both standard, by performing an analysis similar to theone in the proof of Theorem 4.4, and of Theorem 4.9. For full details, see [6]. (cid:3)
UANTUM GROUPS 19
Following now [3], we can formulate a key definition, as follows:
Definition 5.4.
Given a closed subgroup G ⊂ U + N , we set C ( K ) = C ( G ) .D u ij u ∗ ij = u ∗ ij u ij = magic E and we call K ⊂ G the quantum reflection group associated to G . Here the fact that K is indeed a closed subgroup of G comes from the fact that if u = ( u ij ) has the property that its entries are normal, and are such that p ij = u ij u ∗ ij forma magic matrix, then the same holds for u ∆ ij = P k u ik ⊗ u kj , u εij = δ ij , u Sij = u ∗ ji .As basic examples here, for G = O N , U N we obtain the groups K = H N , K N , which arethe hyperoctahedral group H N = Z ≀ S N , and its complex version K N = T ≀ S N .The free analogues of these results are as follows: Proposition 5.5.
The quantum reflection groups associated to O + N , U + N are given by C ( K + N ) = C ( U + N ) .D u ij u ∗ ij = u ∗ ij u ij = magic E and H N = K + N ∩ O N , and we have decompositions H + N = Z ≀ ∗ S + N and K + N = T ≀ ∗ S + N .Proof. The fact that we have indeed decompositions as above is standard, and can bechecked by constructing a pair of inverse isomorphisms, at the algebra level. (cid:3)
More generally now, we have the following result:
Theorem 5.6.
The basic basic unitary quantum groups and the basic twisted unitarygroups have common quantum reflection groups, as follows, K N / / K ∗ N / / K + N H N / / O O H ∗ N / / O O H + N O O where H ∗ N , K ∗ N are obtained from H + N , K + N by intersecting with U ∗ N . These quantum reflec-tion groups are all easy, the corresponding categories of partitions being P even (cid:15) (cid:15) P ∗ even o o (cid:15) (cid:15) N C even o o (cid:15) (cid:15) P even P ∗ even o o N C even o o where P even , P ∗ even , P even are the straightforward multi-block analogues of the categories P , P ∗ , P , and P ∗ even = P even ∩ P ∗ even , N C even = P even ∩ N C , N C even = P even ∩ N C . Proof.
Here the self-duality claim can be proved by using the various formulae in Propo-sition 4.6 above, and the easiness assertion is routine, by proceeding as in the proof ofTheorem 4.4. For details regarding these results, we refer to [1], [2], [6]. (cid:3)
Our claim now is that, in the case of the above quantum groups, G can be in factreconstructed from K . In order to explain this material, we will need: Definition 5.7.
Given two closed subgroups
G, H ⊂ U + N , with fundamental corepresenta-tions denoted u, v , we construct a Tannakian category C = ( C kl ) by setting C kl = Hom ( u ⊗ k , u ⊗ l ) ∩ Hom ( v ⊗ k , v ⊗ l ) and we let < G, H > ⊂ U + N be the associated quantum group. That is, the correspondingfundamental corepresentation w must satisfy Hom ( w ⊗ k , w ⊗ l ) = C kl . Here the fact that C = ( C kl ) is indeed a Tannakian category is clear from definitions. Asfor the notation <, > , this comes from the fact that in the classical case, where G, H ⊂ U N ,the closed subgroup of U N that we obtain is indeed the one generated by G, H .In the easy case we have the following result:
Proposition 5.8.
Assuming that
G, H ⊂ U + N are easy quantum groups, with correspond-ing categories of partitions D, E ⊂ P , we have: (1) G ∩ H is easy, with category of partitions < D, E > . (2) < G, H > is easy, with category of partitions D ∩ E .In addition, the same holds for the twisted easy quantum groups.Proof. All the assertions are clear from the Tannakian correspondence between easy ortwisted easy quantum groups, and categories of partitions. (cid:3)
We can now go back to the quantum reflection groups, and formulate:
Theorem 5.9.
Consider one of the basic easy quantum groups, ˙ O N ⊂ G ⊂ U + N , and let H N ⊂ K ⊂ K + N be its associated quantum reflection group. We have then G = < K, ˙ O N > with the < . > operation being the one from Definition 5.7 above.Proof. This follows indeed by using the criterion in Proposition 5.8 above. (cid:3)
There are many other interesting things that can be said about the reflection groupsconstructed above. We refer here to the survey paper [6], and to [2], [3], [34].
UANTUM GROUPS 21 Classification results
We discuss here various classification questions for the closed subgroups G ⊂ U + N , inthe easy case, and in general. As a first, fundamental result, from [16], we have: Theorem 6.1.
There is only one intermediate easy quantum group O N ⊂ G ⊂ O + N namely the half-classical orthogonal group O ∗ N .Proof. We have to compute the categories of pairings
N C ⊂ D ⊂ P .Step I. Let π ∈ P − N C , having s ≥ π ∈ P − P ∗ , there exists a semicircle capping π ′ ∈ P − P ∗ .(2) If π ∈ P ∗ − N C , there exists a semicircle capping π ′ ∈ P ∗ − N C .Indeed, both these assertions can be easily proved, by drawing pictures.Step II. Consider now a partition π ∈ P ( k, l ) − N C ( k, l ). Our claim is that:(1) If π ∈ P ( k, l ) − P ∗ ( k, l ) then < π > = P .(2) If π ∈ P ∗ ( k, l ) − N C ( k, l ) then < π > = P ∗ .This can be indeed proved by recurrence on the number of strings, s = ( k + l ) /
2, byusing Step I, which provides us with a descent procedure s → s −
1, at any s ≥ O N ⊂ G ⊂ O + N ,coming from certain sets of pairings D ( k, l ) ⊂ P ( k, l ). We have three cases:(1) If D P ∗ , we obtain G = O N .(2) If D ⊂ P , D N C , we obtain G = O ∗ N .(3) If D ⊂ N C , we obtain G = O + N . (cid:3) Regarding now the arbitrary easy quantum groups S N ⊂ G ⊂ O + N , we first have: Theorem 6.2.
The classical and free easy quantum groups are as follows, B + N / / B ′ + N → B ′′ + N / / O + N B N / / d d ■■■■■■■ B ′ N O O / / O N : : ✉✉✉✉✉✉✉✉ S N / / O O z z ✉✉✉✉✉✉✉✉ S ′ N O O (cid:15) (cid:15) / / H N O O $ $ ■■■■■■■ S + N O O / / S ′ + N / / H + N O O where S ′ N = S N × Z , B ′ N = B N × Z , and S ′ + N , B ′ + N , B ′′ + N are their liberations.Proof. The idea here is that of jointly classifying the “classical” categories of partitions P ⊂ D ⊂ P , and the “free” ones N C ⊂ D ⊂ N C . At the classical level this leads to S ′ N , B ′ N . See [15]. At the free level we obtain 3 more quantumgroups, S ′ + N , B ′ + N , B ′′ + N , with the inclusion B ′ + N ⊂ B ′′ + N being best thought of as comingfrom an inclusion B ′ N ⊂ B ′′ N , which happens to be an isomorphism. See [15]. (cid:3) We can complete the above diagram with a number of intermediate liberations. Theconstructions and result here, which are quite technical, are as follows:
Theorem 6.3.
The extra examples of easy quantum groups are as follows: (1)
Half-liberations O N ⊂ O ∗ N ⊂ O ∗ N and H N ⊂ H ∗ N ⊂ H + N and B ′ N ⊂ B ′′∗ N ⊂ B ′′ + N ,obtained by imposing the half-commutation relations abc = cba . (2) A higher half-liberation H ∗ N ⊂ H [ ∞ ] N ⊂ H + N , obtained by imposing the relations abc = 0 , for any a = c on the same row or column of u . (3) An uncountable family of intermediate quantum groups S N ⊂ H Γ N ⊂ H [ ∞ ] N , obtainedfrom the quotients Z ∗∞ → Γ satisfying a certain uniformity condition. (4) A series of intermediate quantum groups H [ ∞ ] N ⊂ H ⋄ kN ⊂ H + N , obtained via therelations [ a . . . a k − b a k − . . . a , c ] = 0 .Proof. The construction and study of O ∗ N go back to [15], [16], the quantum groups H ∗ N , H [ ∞ ] N are from [10], and the constructions of the family H Γ N and of the series H ⋄ kN , aswell as the proof of the classification result, are from [41]. (cid:3) All this is quite technical, and in what follows, our purpose will be just of extendingTheorem 6.1 above. In order to cut a bit from complexity, we will use:
Proposition 6.4.
For a liberation operation of easy quantum groups G N → G + N , thefollowing conditions are equivalent: (1) The category P ⊂ D ⊂ P associated to G = ( G N ) , or, equivalently, the category N C ⊂ D ⊂ N C associated to G + = ( G + N ) , is stable under removing blocks. (2) We have G N ∩ U K = G K , or, equivalently, G + N ∩ U + K = G + K , for any K ≤ N , wherethe embeddings U K ⊂ U N and U + K ⊂ U + N are the standard ones. (3) Each G N appears as lift of its projective version G N → P G N , or, equivalently,each G + N appears as lift of its projective version G + N → P G + N . (4) The laws of truncated characters χ t = P [ tN ] i =1 u ii , with t ∈ (0 , , for G N and G + N ,form convolution/free convolution semigroups, in Bercovici-Pata bijection.If these conditions are satisfied, we call G N → G + N a “true” liberation.Proof. All this is well-known, basically going back to [15], the idea being that the impli-cations (1) ⇐⇒ (2) ⇐⇒ (3) are all elementary, and that (1) ⇐⇒ (4) follows byusing the cumulant interpretation of the Bercovici-Pata bijection [17], stating that “theclassical cumulants become via the bijection free cumulants”. See [14], [15]. (cid:3) We can now extend Theorem 6.1, as follows:
UANTUM GROUPS 23
Theorem 6.5.
There are precisely true liberations of orthogonal easy quantum groups,with the intermediate liberations, in the easy framework, being as follows: (1) S N ⊂ S + N , with no intermediate object. (2) O N ⊂ O + N , with O ∗ N as unique intermediate object. (3) H N ⊂ H + N , with uncountably many intermediate objects. (4) B N ⊂ B + N , with no intermediate object.Proof. The fact that the true liberations are indeed those in the statement follows fromProposition 6.4. As for the other statements, the proof here is routine. See [10]. (cid:3)
One interesting question is that of finding the intermediate quantum groups, not nec-essarily easy, for the above inclusions. There are several open problems here, the mainone being that there is no intermediate quantum group S N ⊂ G ⊂ S + N . See [6].In the unitary case, the situation is considerably more complicated. We first have: Proposition 6.6.
Given S ∈ { , , , . . . , ∞} , we have an intermediate easy quantumgroup O + N ⊂ O + N,S ⊂ O + N, ∞ , obtained by imposing the relations a i . . . a i S = a ∗ i . . . a ∗ i S to the standard coordinates of O + N, ∞ , with the convention that at S = ∞ this relationdissapears. At S = 1 we obtain in this way the quantum group O + N .Proof. The relations in the statement are implemented by the following pairing: ◦ . . . ◦ ( S ) • . . . • But this gives the first assertion, and the last assertion is clear as well. (cid:3)
Now by taking intersections, we are led to the following result:
Theorem 6.7.
We have interesection/generation diagrams of easy quantum groups andof categories of pairings, in Tannakian correspondence, as follows, U N / / U ∗ N / / U + N O N,S / / O O O ∗ N,S / / O O O + N,S O O O N / / O O O ∗ N / / O O O + N O O : P (cid:15) (cid:15) P ∗ o o (cid:15) (cid:15) N C o o (cid:15) (cid:15) P S (cid:15) (cid:15) P S, ∗ o o (cid:15) (cid:15) N C S o o (cid:15) (cid:15) P P ∗ o o N C o o where P S ⊂ P is the set of pairings which, when flattened, have the same number of ◦ , • symbols, modulo S , and where the remaining categories appear as intersections. Proof.
We already know from Theorem 4.4 that the correspondence holds for the upperand lower rows. As for the middle row, the proof here is standard as well.Regarding the intersection/generation claim, which states that any square subdiagram A ⊂ B, C ⊂ D is subject to the conditions A = B ∩ C, D = < B, C > , this is clear for thepairings, and for the quantum groups this follows by using Proposition 5.8. See [2]. (cid:3) The above quantum groups can be characterized as follows:
Theorem 6.8.
The intermediate easy quantum groups O N ⊂ G ⊂ U + N satisfying G = < G class , G real > are precisely those constructed in Theorem 6.7 above.Proof. According to [42], the only easy quantum groups O N ⊂ G ⊂ U N are the compactgroups O N,S , with S ∈ { , , . . . , ∞} . Thus, we must have G class ∈ { O N , O N,S , U N } .On the other hand, we know as well from Theorem 6.1 that we must have G real ∈{ O N , O ∗ N , O + N } . Together with Theorem 6.7 above, this gives the result. (cid:3) The classification in the unitary case, and notably the classification of the intermediateeasy quantum groups O N ⊂ G ⊂ U + N , remains an open problem. See [2], [5], [18].7. Maximal tori
In this section and in the next one we discuss various methods for the study of theclosed subgroups H ⊂ G . The classical subgroups are easy to find, due to: Proposition 7.1.
Given a closed subgroup G ⊂ U + N , the classical subgroups H ⊂ G areprecisely the closed subgroups H ⊂ G class , where G class = G ∩ U N .Proof. This is clear, because the formula G class = G ∩ U N means by definition that wehave C ( G class ) = C ( G ) /J , where J is the commutator ideal of C ( G ). (cid:3) Let us investigate now the group dual subgroups b Λ ⊂ G , whose knowledge is very usefulas well. We have already met, in Proposition 2.9 above, the diagonal torus T ⊂ G . Theconstruction there has the following generalization: Proposition 7.2.
Given a closed subgroup G ⊂ U + N and a matrix Q ∈ U N , we let T Q ⊂ G be the diagonal torus of G , with fundamental representation spinned by Q : C ( T Q ) = C ( G ) . D ( QuQ ∗ ) ij = 0 (cid:12)(cid:12)(cid:12) ∀ i = j E This torus is then a group dual, T Q = b Λ Q , where Λ Q = < g , . . . , g N > is the discrete groupgenerated by the elements g i = ( QuQ ∗ ) ii , which are unitaries inside C ( T Q ) .Proof. This follows indeed from Proposition 2.9 because, as said in the statement, T Q isby definition a diagonal torus, and so is a group dual, as indicated. (cid:3) With this notion in hand, we have the following result, coming from [50]:
UANTUM GROUPS 25
Theorem 7.3.
Given a closed subgroup G ⊂ U + N , its group dual subgroups b Λ ⊂ G areexactly the quantum subgroups of type b Λ ⊂ T Q , with Q ∈ U N .Proof. This follows indeed from Woronowicz’s results in [50]. (cid:3)
Summarizing, if we agree that the group duals are the correct generalization of the“tori” from the classical case, we are led to the following definition:
Definition 7.4.
The maximal torus of a closed subgroup G ⊂ U + N is the family T = (cid:8) T Q ⊂ G (cid:12)(cid:12) Q ∈ U N (cid:9) of diagonal tori of G , parametrized by the various spinning matrices Q ∈ U N . Our aim now is to show that T plays indeed the role of a maximal torus for G . Let usfirst develop some general theory for these maximal tori. We first have: Proposition 7.5.
Given a closed subgroup H ⊂ G , the tori of G, H are related by T Q ( H ) = T Q ( G ) ∩ H with the intersection operation being the usual one, taken inside G .Proof. Let I = ker( C ( G ) → C ( H )). At Q = 1 we have, indeed: C ( T ( H )) = [ C ( G ) /I ] / < u ij = 0 | i = j > = [ C ( G ) / < u ij = 0 | i = j > ] /I = C ( T ( G ) ∩ H )In general, the proof is similar. (cid:3) Let us study the injectivity properties of the construction G → T . We would like forinstance to show that this construction is “strictly increasing” with respect to ⊂ . In otherwords, we would need a result stating that passing to a subgroup H ⊂ G should decreaseat least one of the tori T Q . As a first statement in this direction, we have: Proposition 7.6.
Given a closed subgroup G ⊂ U + N , the following two constructionsproduce the same closed subgroup G ′ ⊂ G : (1) G ′ = < T Q | Q ∈ U N > , the closed subgroup generated by the tori T G ⊂ G . (2) G ′ = < b Λ | b Λ ⊂ G > , the closed subgroup generated by the group duals b Λ ⊂ G .Proof. Let G ′ , G ′ ⊂ G be the two subgroups constructed above. Since any torus T Q isa group dual, we have G ′ ⊂ G ′ . Conversely, since any group dual b Λ ⊂ G appears as asubgroup of a certain torus, b Λ ⊂ T Q ⊂ G , we have G ′ ⊂ G ′ , and we are done. (cid:3) In view of this, it looks reasonable to formulate:
Definition 7.7.
We say that G ⊂ U + N is decomposable when the inclusion G ′ = < T Q | Q ∈ U N > ⊂ G constructed above is an equality, i.e. when G is generated by its tori. At the level of basic examples of such quantum groups, we have:
Theorem 7.8.
The following subgroups G ⊂ U + N are decomposable: (1) The classical groups, G ⊂ U N ⊂ U + N . (2) The group duals, G = b Γ ⊂ U + N .Proof. This is elementary, but not trivial, the proofs being as follows:(1) For G = U N we have T Q ( G ) = Q ∗ T N Q , where T N ⊂ U N are the diagonal matrices,and so by Proposition 7.5 we obtain that for G ⊂ U N we have T Q ( H ) = Q ∗ T N Q ∩ H . Nowsince any group element U ∈ H is diagonalizable, U = Q ∗ DQ with Q ∈ U N , D ∈ T N , wehave U ∈ T Q ( H ) for this value of Q ∈ U N , and this gives the result.(2) This follows from Proposition 7.6 above, or directly from Theorem 7.3. (cid:3) In order to obtain more results, we can use Tannakian duality. We have:
Proposition 7.9.
A closed subgroup G ⊂ U + N is decomposable precisely when ξ ∈ F ix ( u ⊗ kQ ) , ∀ Q ∈ U N = ⇒ ξ ∈ F ix ( u ⊗ k ) where u Q = diag ( g , . . . , g N ) is the fundamental corepresentation of T Q ⊂ G .Proof. The Tannakian category associated to G ′ = < T Q | Q ∈ U N > is given by: Hom ( v ⊗ k , v ⊗ l ) = \ Q ∈ U N Hom ( u ⊗ kQ , u ⊗ lQ )We conclude that the equality G = G ′ is equivalent to the following collection ofconditions, one for any pair k, l of colored integers, as in [36]: Hom ( u ⊗ k , u ⊗ l ) = \ Q ∈ U N Hom ( u ⊗ kQ , u ⊗ lQ )Moreover, by Frobenius duality we can restrict the attention if we want to the spacesof fixed points, and this gives the conclusion in the statement. See [36], [51]. (cid:3) In order to apply the above result, we can use the following formula, from [13]:
Proposition 7.10.
The intertwining formula T ∈ Hom ( u ⊗ k , u ⊗ l ) , with u = QvQ ∗ , where v = diag ( g , . . . , g N ) , is equivalent to the collection of conditions ( T Q ) j ...j l ,i ...i k = 0 = ⇒ g i . . . g i k = g j . . . g j l one for each choice of the multi-indices i, j , where T Q = ( Q ∗ ) ⊗ l T Q ⊗ k . UANTUM GROUPS 27
Proof.
It is enough to prove the result at Q = 1, and here we have: T ∈ Hom ( u ⊗ k , u ⊗ l ) ⇐⇒ X j T ji e j ⊗ g i = X j T ji e j ⊗ g j , ∀ i ⇐⇒ T ji g i = T ji g j , ∀ i, j ⇐⇒ [ T ji = 0 = ⇒ g i = g j ] , ∀ i, j Thus we have obtained the relation in the statement, and we are done. (cid:3)
Now by putting everything together, we obtain:
Theorem 7.11.
A closed subgroup G ⊂ U + N is decomposable precisely when ( T Q ) j ...j l ,i ...i k = 0 = ⇒ g i . . . g i k = g j . . . g j l (inside Γ Q ) for any Q ∈ U N implies T ∈ Hom ( u ⊗ k , u ⊗ l ) , and this, for any k, l .Proof. This follows indeed from Proposition 7.9 and Proposition 7.10 above. (cid:3)
The above result is of course something quite technical, rather waiting to be applied invarious concrete situations. In general, understanding the structure of the decomposablesubgroups G ⊂ U + N , and the injectivity properties of the maximal torus construction G → T , are definitely interesting problems, that we would like to raise here.There are as well several explicit conjectures regarding the maximal tori, the generalidea being that the knowledge of T solves most of the problems regarding G . See [13].8. Matrix models
In this section we discuss matrix modelling questions for the closed subgroups G ⊂ U + N .We use the following matrix model formalism: Definition 8.1.
A matrix model for C ( G ) is a morphism of C ∗ -algebras π : C ( G ) → M K ( C ( X )) with X being a compact space, and with K ∈ N . The “best” kind of models are of course the faithful ones. However, since having anembedding C ( G ) ⊂ M K ( C ( X )) forces the algebra C ( G ) to be of type I, and so G to becoamenable, we cannot expect such models to exist, in general. For instance O + N with N ≥
3, or U + N with N ≥
2, which are not coamenable, cannot have such models.However, we have some interesting constructions of faithful models, as follows:
Theorem 8.2.
The quantum groups O ∗ N , U ∗ N are as follows: (1) We have an embedding C ( O ∗ N ) ⊂ M ( C ( U N )) , mapping the coordinates u ij toantidiagonal self-adjoint matrices. (2) We have as well an embedding C ( U ∗ N ) ⊂ M ( C ( U N × U N )) , obtained by usingantidiagonal unitary matrices. Proof.
Here the fact that we have a morphisms as in (1,2) is clear, because the antidi-agonal matrices half-commute. The faithfulness of these models can be proved by usingrepresentation theory methods, and more specifically Theorem 4.4. See [5], [7], [19]. (cid:3)
In order to deal now with the non-amenable case, let us go back to Definition 8.1. Inthe group dual case G = b Γ, a matrix model π : C ∗ (Γ) → M K ( C ( X )) must come from agroup representation ρ : Γ → C ( X, U K ). Now observe that when ρ is faithful, the inducedrepresentation π is in general not faithful, its target algebra being finite dimensional. Onthe other hand, this representation “reminds” Γ. We say that π is inner faithful.We have in fact the following notions, coming from [4]: Definition 8.3.
Let π : C ( G ) → M K ( C ( X )) be a matrix model. (1) The Hopf image of π is the smallest quotient Hopf C ∗ -algebra C ( G ) → C ( H ) producing a factorization of type π : C ( G ) → C ( H ) → M K ( C ( X )) . (2) When the inclusion H ⊂ G is an isomorphism, i.e. when there is no non-trivialfactorization as above, we say that π is inner faithful. In the case where G = b Γ is a group dual, π must come from a group representation ρ : Γ → C ( X, U K ), and the above factorization is simply the one obtained by taking theimage, ρ : Γ → Γ ′ ⊂ C ( X, U K ). Thus π is inner faithful when Γ ⊂ C ( X, U K ).Also, given a compact group G , and elements g , . . . , g K ∈ G , we have a representation π : C ( G ) → C K , given by f → ( f ( g ) , . . . , f ( g K )). The minimal factorization of π is thenvia C ( G ′ ), with G ′ = < g , . . . , g K > , and π is inner faithful when G = G ′ .We refer to [4], [20], [22] for more on these facts, and for a number of related algebraicresults. In what follows, we will rather use analytic techniques. Assume indeed that X isa probability space. We have then the following result, from [11], [48]: Theorem 8.4.
Given an inner faithful model π : C ( G ) → M K ( C ( X )) , we have Z G = lim k →∞ k k X r =1 Z rG where R rG = ( ϕ ◦ π ) ∗ r , with ϕ = tr ⊗ R X being the random matrix trace.Proof. This was proved in [11] in the case X = { . } , using idempotent state theory from[28]. The general case was recently established in [48]. (cid:3) The truncated integrals R rG can be evaluated as follows: Proposition 8.5.
Assuming that π : C ( G ) → M K ( C ( X )) maps u ij → U xij , we have Z rG u e i j . . . u e p i p j p = ( T re ) i ...i p ,j ...j p where T e ∈ M N p ( C ) with e ∈ { , ∗} p is given by ( T e ) i ...i p ,j ...j p = R X tr ( U x,e i j . . . U x,e p i p j p ) dx . UANTUM GROUPS 29
Proof.
This follows indeed from the definition of the various objects involved, namelyfrom φ ∗ ψ = ( φ ⊗ ψ )∆, and from ∆( u ij ) = P k u ik ⊗ u kj . See [5]. (cid:3) As a first application, we can further investigate the faithful models, by using:
Definition 8.6.
A stationary model for C ( G ) is a random matrix model π : C ( G ) → M K ( C ( X )) having the property R G = ( tr ⊗ R X ) π . Observe that any stationary model is faithful. Indeed, the stationarity condition givesa factorization π : C ( G ) → C ( G ) red ⊂ M K ( C ( X )), and since the image algebra C ( G ) red follows to be of type I, and therefore nuclear, G must be co-amenable, and so π must befaithful. For some background on these questions, we refer to [38].As a useful criterion for the stationarity property, we have: Proposition 8.7.
For π : C ( G ) → M K ( C ( X )) , the following are equivalent: (1) Im ( π ) is a Hopf algebra, and ( tr ⊗ R X ) π is the Haar integration on it. (2) ψ = ( tr ⊗ R X ) π satisfies the idempotent state property ψ ∗ ψ = ψ . (3) T e = T e , ∀ p ∈ N , ∀ e ∈ { , ∗} p , where ( T e ) i ...i p ,j ...j p = ( tr ⊗ R X )( U e i j . . . U e p i p j p ) .If these conditions are satisfied, we say that π is stationary on its image.Proof. Let us factorize our matrix model, as in Definition 8.3 above: π : C ( G ) → C ( G ′ ) → M K ( C ( X ))Now observe that the conditions (1,2,3) only depend on the factorized representation π ′ : C ( G ′ ) → M K ( C ( X )). Thus, we can assume G = G ′ , which means that we can assumethat π is inner faithful. We can therefore use the formula in Theorem 8.4: Z G = lim k →∞ k k X r =1 ψ ∗ r (1) = ⇒ (2) This is clear from definitions, because the Haar integration on any quantumgroup satisfies the equation ψ ∗ ψ = ψ .(2) = ⇒ (1) Assuming ψ ∗ ψ = ψ , we have ψ ∗ r = ψ for any r ∈ N , and the above Ces`arolimiting formula gives R G = ψ . By using now the amenability arguments explained afterDefinition 8.6, we obtain as well that π is faithful, as desired.In order to establish now (2) ⇐⇒ (3), we use the formula in Proposition 8.5: ψ ∗ r ( u e i j . . . u e p i p j p ) = ( T re ) i ...i p ,j ...j p (2) = ⇒ (3) Assuming ψ ∗ ψ = ψ , by using the above formula at r = 1 , T e and T e have the same coefficients, and so they are equal.(3) = ⇒ (2) Assuming T e = T e , by using the above formula at r = 1 , ψ and ψ ∗ ψ coincide on any product of coefficients u e i j . . . u e p i p j p . Nowsince these coefficients span a dense subalgebra of C ( G ), this gives the result. (cid:3) As a basic application of the above result, we have:
Theorem 8.8.
The standard matrix models for the algebras C ( O ∗ N ) , C ( U ∗ N ) , constructedin Theorem 8.2 above, are stationary.Proof. This follows indeed from a routine Haar measure computation, which enhances thealgebraic considerations from the proof of Theorem 8.2. See [7]. (cid:3)
There are many other interesting examples of stationary models, including the Paulimatrix model for the algebra C ( S +4 ), discussed in [6]. We refer to [7] and to subsequentpapers for more on this subject, and for some recent results on the non-stationary case aswell. There might be actually a relation here with lattice models too [27].Finally, many interesting questions arise in relation with Connes’ noncommutative ge-ometry [24], and we refer here to [18], [23], [26], [31]. Also, we refer to [20], [22], [43],[44] for more specialized analytic aspects, and to [32] and subsequent papers for free deFinetti theorems [32], in the spirit of Voiculescu’s free probability theory [45]. References [1] T. Banica, Liberations and twists of real and complex spheres,
J. Geom. Phys. (2015), 1–25.[2] T. Banica, Unitary easy quantum groups: geometric aspects, J. Geom. Phys. (2018), 127–147.[3] T. Banica, S.T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws,
Canad. J. Math. (2011),3–37.[4] T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. (2010),677–703.[5] T. Banica and J. Bichon, Matrix models for noncommutative algebraic manifolds, J. Lond. Math.Soc. (2017), 519–540.[6] T. Banica, J. Bichon and B. Collins, Quantum permutation groups: a survey, Banach Center Publ. (2007), 13–34.[7] T. Banica and A. Chirvasitu, Thoma type results for discrete quantum groups, Internat. J. Math. (2017), 1–23.[8] T. Banica and B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. (2007), 277–302.[9] T. Banica and S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. (2010), 1–14.[10] T. Banica, S. Curran and R. Speicher, Classification results for easy quantum groups, Pacific J.Math. (2010), 1–26.[11] T. Banica, U. Franz and A. Skalski, Idempotent states and the inner linearity property,
Bull. Pol.Acad. Sci. Math. (2012), 123–132.[12] T. Banica and I. Nechita, Block-modified Wishart matrices and free Poisson laws, Houston J. Math. (2015), 113–134.[13] T. Banica and I. Patri, Maximal torus theory for compact quantum groups, Illinois J. Math. (2017), 151–170.[14] T. Banica, A. Skalski and P.M. So ltan, Noncommutative homogeneous spaces: the matrix case, J.Geom. Phys. (2012), 1451–1466.[15] T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. (2009), 1461–1501.
UANTUM GROUPS 31 [16] T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group,
Ann. Inst. Fourier (2010), 2137–2164.[17] H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory, Ann. ofMath. (1999), 1023–1060.[18] J. Bhowmick, F. D’Andrea and L. Dabrowski, Quantum isometries of the finite noncommutativegeometry of the standard model,
Comm. Math. Phys. (2011), 101–131.[19] J. Bichon and M. Dubois-Violette, Half-commutative orthogonal Hopf algebras,
Pacific J. Math. (2013), 13–28.[20] M. Brannan, B. Collins and R. Vergnioux, The Connes embedding property for quantum group vonNeumann algebras,
Trans. Amer. Math. Soc. (2017), 3799–3819.[21] R. Brauer, On algebras which are connected with the semisimple continuous groups,
Ann. of Math. (1937), 857–872.[22] A. Chirvasitu, Residually finite quantum group algebras, J. Funct. Anal. (2015), 3508–3533.[23] F. Cipriani, U. Franz and A. Kula, Symmetries of L´evy processes on compact quantum groups, theirMarkov semigroups and potential theory,
J. Funct. Anal. (2014), 2789–2844.[24] A. Connes, Noncommutative geometry, Academic Press (1994).[25] A. D’Andrea, C. Pinzari and S. Rossi, Polynomial growth for compact quantum groups, topologicaldimension and *-regularity of the Fourier algebra, preprint 2016.[26] B. Das and D. Goswami, Quantum Brownian motion on noncommutative manifolds: construction,deformation and exit times,
Comm. Math. Phys. (2012), 193–228.[27] L. Faddeev, Instructive history of the quantum inverse scattering method,
Acta Appl. Math. (1995), 69–84.[28] U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra (2009), 1774–1802.[29] A. Freslon, On the partition approach to Schur-Weyl duality and free quantum groups,
Transform.Groups (2017), 707–751.[30] M. Fukuda and P. ´Sniady, Partial transpose of random quantum states: exact formulas and meanders, J. Math. Phys. (2013), 1–31.[31] D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math.Phys. (2009), 141–160.[32] C. K¨ostler, R. Speicher, A noncommutative de Finetti theorem: invariance under quantum permu-tations is equivalent to freeness with amalgamation,
Comm. Math. Phys. (2009), 473–490.[33] J. Kustermans and S. Vaes, Locally compact quantum groups,
Ann. Sci. Ecole Norm. Sup. (2000),837–934.[34] F. Lemeux and P. Tarrago, Free wreath product quantum groups: the monoidal category, approxi-mation properties and free probability, J. Funct. Anal. (2016), 3828–3883.[35] A. Maes and A. Van Daele, Notes on compact quantum groups,
Nieuw Arch. Wisk. (1998),73–112.[36] S. Malacarne, Woronowicz’s Tannaka-Krein duality and free orthogonal quantum groups, Math.Scand. (2018), 151–160.[37] J.A. Mingo and M. Popa, Freeness and the partial transposes of Wishart random matrices, preprint2017.[38] S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories, SMF(2013).[39] G.K. Pedersen, C ∗ -algebras and their automorphism groups, Academic Press (1979).[40] P. Podle´s and S.L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. (1990), 381–431. [41] S. Raum and M. Weber, The full classification of orthogonal easy quantum groups,
Comm. Math.Phys. (2016), 751–779.[42] P. Tarrago and M. Weber, Unitary easy quantum groups: the free case and the group case, preprint2015.[43] S. Vaes and R. Vergnioux, The boundary of universal discrete quantum groups, exactness and fac-toriality,
Duke Math. J. (2007), 35–84.[44] R. Vergnioux and C. Voigt, The K-theory of free quantum groups,
Math. Ann. (2013), 355–400.[45] D.V. Voiculescu, K.J. Dykema and A. Nica, Free random variables, AMS (1992).[46] S. Wang, Free products of compact quantum groups,
Comm. Math. Phys. (1995), 671–692.[47] S. Wang, Quantum symmetry groups of finite spaces,
Comm. Math. Phys. (1998), 195–211.[48] S. Wang, L p -improving convolution operators on finite quantum groups, Indiana Univ. Math. J. (2016), 1609–1637.[49] H. Wenzl, C ∗ -tensor categories from quantum groups, J. Amer. Math. Soc. (1998), 261–282.[50] S.L. Woronowicz, Twisted SU (2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. (1987), 117–181.[51] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. (1987), 613–665.[52] S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups,