## Representations of compact quantum groups and subfactors (1999)

Venue: | J. Reine Angew. Math |

Citations: | 34 - 18 self |

### BibTeX

@ARTICLE{Banica99representationsof,

author = {Teodor Banica},

title = {Representations of compact quantum groups and subfactors},

journal = {J. Reine Angew. Math},

year = {1999}

}

### OpenURL

### Abstract

Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. This is related with our previous work [B2]. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.

### Citations

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Free random variables
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Citation Context ...n A ∗red B is also a Woronowicz algebra (which is equal to the reduced version of the Woronowicz algebra A ∗ B). The C ∗ -algebras Ared and Bred are embedded in A ∗red B, and are free in the sense of =-=[VDN]-=- with respect to h ∗ k. Consider the Woronowicz algebra C ∗ (Z) and let z be the unitary of C ∗ (Z) corresponding to the generator 1 of Z ; it is a one-dimensional corepresentation of C ∗ (Z). Recall ... |

263 |
Compact matrix pseudogroups
- Woronowicz
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Citation Context ...ts of the form L(π), with π a representation of a compact group G (resp. of a “dual” ̂ Γ of a discrete group). These objects Gq (q > 0), G and ̂Γ are compact quantum groups in the sense of Woronowicz =-=[W1]-=-,[W2],[W3] (for Gq, see [R2]), and in fact the following general result holds. Theorem A. If π is a finite dimensional unitary representation of a compact quantum group G, then the lattice L(π) is a P... |

133 |
Unitaires multiplicatifs et dualité pour les produits croisés de C∗- algèbres
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- 1993
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Citation Context ... we have written this introductory section by using the more suggestive formalism of quantum groups associated with Woronowicz algebras. It has to be mentioned that the locally compact quantum groups =-=[BS]-=- are known to be related to the depth 2 subfactors by a crossed product construction (see [EN] and the references therein). The point of view in this paper is completely different - in fact we relate ... |

118 |
Hecke algebras of type An and subfactors
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Citation Context ...was intensively investigated for the quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see =-=[We1]-=-,[We2],[We3],[X] and also [J1],[TL]. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [X]. Also the subfactors constructed by Wasse... |

109 |
A new duality theory for compact groups
- Doplicher, Roberts
- 1989
(Show Context)
Citation Context ...ication of fiber functors on R(Aij)0≤i≤j<∞ . While this can be done in certain cases, in general this is related to an unsolved problem, namely the generalisation of the theorems of Doplicher-Roberts =-=[DR]-=- and Deligne [D]. The paper is organised as follows. In the first section we recall Woronowicz’ formalism, we discuss the notion of duality for unitary representations, and we give a list of relevant ... |

90 |
Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent
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Citation Context ... The q-deformation of the C ∗ -algebras C(G), with G compact classical Lie group and q > 0 was started by Woronowicz, and completed by Rosso [R2], via a beautiful application of the Tannakian duality =-=[W2]-=-. By [R1],[R2] the fusion semiring is invariant under q-deformations. If µ ∈ [−1, 1] − {0} and u is the fundamental representation of SµU(2) then Qu = diag(| µ | −1 , | µ |) (cf. the Appendix of [W1])... |

80 |
Entropy and index for subfactors
- Pimsner, Popa
- 1986
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Citation Context ... ). In fact these are (modulo some “unitary equivalence”, see below) all the representations of (A λ ij )0≤i≤j<∞ on finite dimensional Hilbert spaces. Note that the so-called PPTL representation (see =-=[16]-=- and section 5 in [10]) is a particular case of this construction. The second example is in fact a particular case of the first one, for v = the fundamental representation of Au(Q). In fact the follow... |

66 | Operator algebras and conformal field theory III: fusion of positive representations of LSU(n) using bounded operators
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- 1998
(Show Context)
Citation Context ...he quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see [30],[31],[32],[36] and also [11],=-=[29]-=-,[24]. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [36]. Also Wassermann’s subfactors associated to representations of compact... |

58 |
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(Show Context)
Citation Context ... if ( ˜ G, ˜π) is the universal pair satisfying L(˜π) = L(π) as sublattices of L(H) then there exists an (explicit) embedding ̂ G˜ ֒→ Z ∗ π(G). ̂ Here ∗ is the free product of discrete quantum groups =-=[Wn]-=- (see the section 5 for the rigorous statement, in terms of Hopf algebras). The proof uses an isomorphism criterion which relies on a result of free probability theory of Nica and Speicher [NS] and on... |

55 |
gories Tannakiennes
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- 1991
(Show Context)
Citation Context ...functors on R(Aij)0≤i≤j<∞ . While this can be done in certain cases, in general this is related to an unsolved problem, namely the generalisation of the theorems of Doplicher-Roberts [DR] and Deligne =-=[D]-=-. The paper is organised as follows. In the first section we recall Woronowicz’ formalism, we discuss the notion of duality for unitary representations, and we give a list of relevant examples. In the... |

51 |
Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra
- Rosso
- 1988
(Show Context)
Citation Context ...formation of the C ∗ -algebras C(G), with G compact classical Lie group and q > 0 was started by Woronowicz, and completed by Rosso [R2], via a beautiful application of the Tannakian duality [W2]. By =-=[R1]-=-,[R2] the fusion semiring is invariant under q-deformations. If µ ∈ [−1, 1] − {0} and u is the fundamental representation of SµU(2) then Qu = diag(| µ | −1 , | µ |) (cf. the Appendix of [W1]). Example... |

48 |
Algebraic Aspects of the Quantum Yang-Baxter Equation
- Gurevich
- 1991
(Show Context)
Citation Context ...le for the considerations in the end of the Introduction. As a matter of fact, we mention that the Woronowicz algebras Ao(F) are exactly the “compact forms” of the Gurevich’s quantizations of SL(2,C) =-=[G]-=-. Example 1.4. For n ∈ N and F ∈ GL(n,C) let Au(F) be the universal C ∗ -algebra generated by the entries of a n × n matrix u with the relations u unitary, FuF −1 unitary ; these Woronowicz algebras c... |

47 | Compact quantum groups - Woronowicz - 1998 |

39 |
Markov traces on universal Jones algebras and subfactors of finite index
- Popa
- 1993
(Show Context)
Citation Context ...,j] ) being generated by the Jones projections (cf. the representation theory of SµU(2), see [W2]), End(v ⊗[i,j] )0≤i≤j<∞ is the lattice of higher relative commutants of the subfactors constructed in =-=[P2]-=-. 153 Representations of Popa systems There are several restrictions on the Popa systems associated to corepresentations of Woronowicz algebras, for instance the index is always ≥ 4 and the index of ... |

39 |
Universal quantum groups
- Daele, Wang
- 1996
(Show Context)
Citation Context ... F ∈ GL(n,C) satisfying FF ∈ RIn let Ao(F) be the universal C ∗ -algebra generated by the entries of a n ×n matrix u with the relations u = FuF −1 = unitary. This Woronowicz algebra was introduced in =-=[VDW]-=- as to represent the “free analogue of O(n)” and its fusion semiring was shown in [B1] to be isomorphic to the one of SU(2). Moreover, by combining this result with [KW] one finds that the category of... |

38 |
C*-Tensor categories from quantum groups
- Wenzl
- 1998
(Show Context)
Citation Context ...ely investigated for the quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see [We1],[We2],=-=[We3]-=-,[X] and also [J1],[TL]. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [X]. Also the subfactors constructed by Wassermann in [Ws... |

37 |
Déformations de C ∗ -algèbres de
- Blanchard
- 1996
(Show Context)
Citation Context ...[B2] on Au(F) which extend to the algebras of the form C( ˜ G). The last result is about amenability. For locally compact quantum groups the basic results on amenability were established by Blanchard =-=[Bl]-=-, and in the discrete case a Kesten type result could be deduced from his work (this was explained to us by G. Skandalis). This result has several applications (see the section 6), one of them being: ... |

33 |
Le groupe quantique compact libre
- BANICA
- 1997
(Show Context)
Citation Context ...iece of FG”. Thus what we have to do for proving (ii) =⇒ (i) is to construct a monoidal category and a monoidal functor when knowing “pieces” of them; and this kind of problem (see also [34],[13],[2],=-=[3]-=-) is well-known to be usually a combinatorial one. The construction is not unique, and in fact we find the “universal” pair (G, π) such that (Aij)0≤i≤j<∞ is equal to L(π) (as sublattices of L(H)). The... |

31 |
Théorie des représentations du groupe quantique compact libre O(n
- BANICA
- 1996
(Show Context)
Citation Context ...ations of (composable) compositions of tensor products of maps of the form iα, iβ, pα, pβ, or of the form idx := idH ⊗x with x ∈ N∗2 , or of the form T with T ∈ Ax for some x ∈ N∗2 alt (see also [34],=-=[2]-=-,[3] for this kind of constructions). By definition of R we have Ax ⊂ EndR(H ⊗x ) for any x ∈ N∗2 alt. Theorem 3.1 will follow from the reconstruction results in [34] and from the following result. Pr... |

30 |
An axiomatization of the lattice of higher relative commutants of a subfactor
- Popa
- 1995
(Show Context)
Citation Context ...nica Institut de Mathématiques de Luminy, case 930, F-13288 Marseille Cedex 9, France E-mail: banica@iml.univ-mrs.fr Abstract: We associate Popa systems (= standard invariants of subfactors, cf. [P3],=-=[P4]-=-) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented” on finite dimensional Hilbert spac... |

27 |
Reconstructing monoidal categories
- Kazhdan, Wenzl
- 1993
(Show Context)
Citation Context ...ponding piece of FG”. Thus what we have to do for proving (ii) =⇒ (i) is to construct a monoidal category and a monoidal functor when knowing “pieces” of them; and this kind of problem (see also [W2],=-=[KW]-=-,[B1],[B2]) is well-known to be usually a combinatorial one. The construction is not unique, and in fact we find the “universal” pair (G, π) such that (Aij)0≤i≤j<∞ is equal to L(π) (as sublattices of ... |

26 |
Irreducible inclusions of factors, multiplicative unitaries and Kac algebras
- Enock, Nest
- 1996
(Show Context)
Citation Context ...roups associated with Woronowicz algebras. It has to be mentioned that the locally compact quantum groups [BS] are known to be related to the depth 2 subfactors by a crossed product construction (see =-=[EN]-=- and the references therein). The point of view in this paper is completely different - in fact we relate the “theory of a single representation of compact quantum groups” with the “theory of the stan... |

24 |
higher relative commutants and the fusion algebra associated to a subfactor, from: “Operator algebras and their applications
- Bisch, Bimodules
- 1994
(Show Context)
Citation Context ... = πk,l(Ak,l). Proof. By a recurrence argument, it is enough to prove it for k = i + 2 and l = j + 2. Recall that the canonical isomorphism sh : Aij → Ai+2,j+2 satisfies the formula (see for instance =-=[Bi]-=-) sh(T)ei+2 = λ i−j ei+2ei+3...ej+1Tej+2ej+1...ei+3ei+2 (⋆) Now by Prop. 3.2 (i,ii) we get that πi,j+2(ei+2...ej+1) = λ j−i/2 iˆγ ⊗ id[i,j−1] ⊗ pδ ⊗ idˆ δ πi,j+2(ej+2...ei+2) = λ j−i+1/2 pγ ⊗ id[i,j] ... |

20 | R-diagonal pairs | a common approach to Haar unitaries and circular elements
- Nica, Speicher
- 1997
(Show Context)
Citation Context ...groups [Wn] (see the section 5 for the rigorous statement, in terms of Hopf algebras). The proof uses an isomorphism criterion which relies on a result of free probability theory of Nica and Speicher =-=[NS]-=- and on the following idea from [B2]: the dimensions of the linear spaces Hom(r, p) with r, p = tensor products between π and ˆπ are exactly the ∗-moments of the character χ(π) ∈ C(G) with respect to ... |

20 |
Coactions and Yang-Baxter equations for ergodic actions and subfactors, in Operator algebras and applications
- Wassermann
- 1988
(Show Context)
Citation Context ...nstructions of new quantum groups and subfactors. Let us mention that for the “classical” compact quantum groups (compact groups, duals of discrete groups, q-deformations) we obtain the subfactors in =-=[Ws]-=-,[P1], respectively those in [X] with q > 0, togehter with some of their properties (e.g. the characterisation of the amenable 1ones). We also mention that when performing the above three operations ... |

18 |
Classification of amenable subfactors of type II
- Popa
- 1994
(Show Context)
Citation Context ...or Banica Institut de Mathématiques de Luminy, case 930, F-13288 Marseille Cedex 9, France E-mail: banica@iml.univ-mrs.fr Abstract: We associate Popa systems (= standard invariants of subfactors, cf. =-=[P3]-=-,[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented” on finite dimensional Hilbert... |

16 |
Fusion of Positive Energy Representations of LSpin2n
- Laredo
- 1997
(Show Context)
Citation Context ... quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see [We1],[We2],[We3],[X] and also [J1],=-=[TL]-=-. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [X]. Also the subfactors constructed by Wassermann in [Ws] and by Popa in [P1] h... |

16 |
Subfactors and knots
- Jones
- 1991
(Show Context)
Citation Context ...(modulo some “unitary equivalence”, see below) all the representations of (A λ ij )0≤i≤j<∞ on finite dimensional Hilbert spaces. Note that the so-called PPTL representation (see [16] and section 5 in =-=[10]-=-) is a particular case of this construction. The second example is in fact a particular case of the first one, for v = the fundamental representation of Au(Q). In fact the following result holds: Theo... |

15 |
Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif
- ROSSO
- 1990
(Show Context)
Citation Context ...a representation of a compact group G (resp. of a “dual” ̂ Γ of a discrete group). These objects Gq (q > 0), G and ̂Γ are compact quantum groups in the sense of Woronowicz [W1],[W2],[W3] (for Gq, see =-=[R2]-=-), and in fact the following general result holds. Theorem A. If π is a finite dimensional unitary representation of a compact quantum group G, then the lattice L(π) is a Popa system. That is, L(π) ha... |

13 | On the Haar measure of the quantum - Nagy - 1993 |

12 |
Fusion en algebres de von Neumann et groupes de lacets [d’apres A. Wassermann]. Seminaire Bourbaki 800
- Jones
- 1995
(Show Context)
Citation Context ...r the quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see [We1],[We2],[We3],[X] and also =-=[J1]-=-,[TL]. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [X]. Also the subfactors constructed by Wassermann in [Ws] and by Popa in [... |

6 |
Sous-facteurs, actions des groupes et cohomologie
- Popa
- 1989
(Show Context)
Citation Context ...of a deformation Gq of a compact group (with q > 0) [36]. Also Wassermann’s subfactors associated to representations of compact groups ([28]) and the locally trivial subfactors of Jones and Popa (see =-=[17]-=-) have higher relative commutants of the form L(π), with π a representation of a compact group G (resp. of a “dual” ̂ Γ of a discrete group). These objects Gq (q > 0), G and ̂Γ are compact quantum gro... |

5 |
Le groupe quantique libre
- Banica
- 1997
(Show Context)
Citation Context ...terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. This is related with our previous work =-=[B2]-=-. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to compare it with the amenability of subfactors. Introduction After Jones’ discovery of the index of ... |

5 |
λ-lattices from quantum groups. preprint Teodor Banica Institut de Mathématiques de Luminy
- Xu, Standard
(Show Context)
Citation Context ... and subfactors. Let us mention that for the “classical” compact quantum groups (compact groups, duals of discrete groups, q-deformations) we obtain the subfactors in [Ws],[P1], respectively those in =-=[X]-=- with q > 0, togehter with some of their properties (e.g. the characterisation of the amenable 1ones). We also mention that when performing the above three operations with the simplest data, namely w... |

4 |
Antony Actions of compact Lie groups on von Neumann algebras
- opa, Wassermann
(Show Context)
Citation Context ...s the corepresentation diag(ug1, ..., ugn) of the Woronowicz algebra A = C ∗ (Γ). It is possible to extend the above results to coactions/actions of more general Woronowicz algebras (see for instance =-=[PW]-=-), and in fact this quantum group formalism shows that the above two constructions are of the same nature (i.e. one can pass from each of them to the other one by considering dual coactions/actions ; ... |

1 |
Quantum groups and subfactors of type B,C
- Wenzl
- 1990
(Show Context)
Citation Context ...ntensively investigated for the quantum groups at roots of unity (which of course are not compact quantum groups), and 2a whole series of interesting subfactors was constructed in this way, see [30],=-=[31]-=-,[32],[36] and also [11],[29],[24]. The above general statement holds also when π is a representation of a deformation Gq of a compact group (with q > 0) [36]. Also Wassermann’s subfactors associated ... |