## Representations of algebraic quantum groups and reconstruction theorems for tensor categories

Citations: | 8 - 4 self |

### BibTeX

@MISC{Müger_representationsof,

author = {M. Müger and J. E. Roberts and L. Tuset},

title = {Representations of algebraic quantum groups and reconstruction theorems for tensor categories},

year = {}

}

### OpenURL

### Abstract

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor ∗-category with conjugates is equivalent to the category of finite dimensional non-degenerate ∗-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka-Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf ∗-algebras. 1

### Citations

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289 | Quantum Groups and their Representations - Klimyk, Schmudgen - 1997 |

265 |
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(Show Context)
Citation Context ...roduction Pontryagin’s duality theory for locally compact abelian groups and the Tannaka-Krein theory for compact groups are two major results in the theory of harmonic analysis on topological groups =-=[14]-=-. The Pontryagin duality theorem is a statement concerning characters, whereas the Tannaka-Krein theorem is a statement involving irreducible unitary representations. These two notions coincide whenev... |

127 |
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Citation Context ... emphasis on ∗-categories. In contradistinction to these authors we wish to distinguish categorical from quantum group aspects as well as algebraic from analytic aspects as far as possible. Following =-=[31, 7, 34, 45]-=-, we emphasize natural transformations of the embedding functor. Yet, our use of natural transformations is more direct and we work with representations rather than corepresentations. The algebra prod... |

121 | Hopf algebras - Abe - 2004 |

118 |
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(Show Context)
Citation Context ...e two notions coincide whenever the group is both abelian and compact. Pontryagin’s theorem can be stated more generally for locally compact quantum groups [21], a notion evolving out of Kac algebras =-=[11]-=- and compact matrix pseudogroups [40]. The theory of representations is most naturally developed in the language of tensor categories [24]. The category of finite dimensional representations of a comp... |

104 |
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(Show Context)
Citation Context ...entations is most naturally developed in the language of tensor categories [24]. The category of finite dimensional representations of a compact group is a symmetric tensor ∗-category with conjugates =-=[9]-=-, and the Tannaka-Krein theorem tells us how to reconstruct the group from the latter. In 1988, starting from a tensor ∗-category [12] with conjugates and admitting a generator, but assuming neither a... |

79 |
Quantum Deformation of Lorentz Group
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- 1990
(Show Context)
Citation Context ...ago. At the time, no appropriate self-dual category of quantum groups existed. (In the same year, P. Podlés and Woronowicz, in defining a discrete quantum group, took the first step in this direction =-=[28]-=-.) The approach to Tannakian categories motivated by algebraic geometry is well reviewed in [6], where, however, only symmetric tensor categories are considered. Our approach to generalized Tannaka-Kr... |

78 |
2003 Locally compact quantum groups
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Citation Context ...ng irreducible unitary representations. These two notions coincide whenever the group is both abelian and compact. Pontryagin’s theorem can be stated more generally for locally compact quantum groups =-=[21]-=-, a notion evolving out of Kac algebras [11] and compact matrix pseudogroups [40]. The theory of representations is most naturally developed in the language of tensor categories [24]. The category of ... |

67 |
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(Show Context)
Citation Context ...sformations of the embedding functor was generalized in the work of K.-H. Ulbrich [34], where a given concrete tensor category is identified as the category of comodules over a Hopf algebra, cf. also =-=[45]-=-. 1While the role of the natural transformations of the embedding functor is obscure in Woronowicz’s approach, they appear at least implicitly in the work of S. Yamagami [43], who considered represen... |

66 |
S.: Locally compact quantum groups in the von Neumann algebraic setting
- Kustermans, Vaes
(Show Context)
Citation Context ...ctional. All compact and all discrete quantum groups are aqg, and every aqg has a unique analytic extension to a locally compact quantum group [20], having an equivalent von Neumann algebraic version =-=[22]-=-. A discrete multiplier Hopf ∗-algebra [35] can be shown to have a Haar functional rendering it a discrete aqg. The purpose of the present paper is to give a coherent and reasonably complete survey of... |

61 |
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Citation Context ... r = ∑ vi ⊗ wi ◦ ri. i We define the (intrinsic or categorical) dimension d(X) ∈ R+ of X by r ∗ ◦ r = d(X)id1 where (X, r, r) is a normalized standard solution. One can prove the following facts, cf. =-=[23]-=-. The dimension is additive under direct sums and multiplicative under tensor products. It takes values in the set {2 cos π n , n = 3, 4, . . . } ∪ [2, ∞), in particular d(X) ≥ 1 with d(X) = 1 iff X ⊗... |

54 |
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Citation Context ...iori. For symmetric categories, it was first shown by S. Doplicher and J. E. Roberts [9] that such a functor always exists. An alternative approach in a more algebraic setting was given by P. Deligne =-=[8]-=-. The questions of existence and uniqueness of an embedding functor will be addressed anew in a sequel to this paper. The above discussion did not follow the order of presentation. Let us therefore gi... |

52 | From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories - Müger |

49 |
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Citation Context ...ndering it a discrete aqg. The purpose of the present paper is to give a coherent and reasonably complete survey of the TannakaKrein theory of quantum groups. The only other review we are aware of is =-=[15]-=-, which appeared ten years ago. At the time, no appropriate self-dual category of quantum groups existed. (In the same year, P. Podlés and Woronowicz, in defining a discrete quantum group, took the fi... |

43 | Analysis Now - Pedersen - 1989 |

42 |
Daele, An Algebraic Framework for Group Duality
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- 1998
(Show Context)
Citation Context ...al case. Motivated by the desire to find a purely algebraic framework for quantum groups admitting a version of Pontryagin duality, A. Van Daele developed the theory of algebraic quantum groups (aqg) =-=[37]-=-. This is achieved by admitting non-unital algebras and requiring the existence of a positive and leftinvariant functional, the Haar functional. All compact and all discrete quantum groups are aqg, an... |

35 | Daele: Multiplier Hopf algebras - Van - 1994 |

29 | Galois theory for braided tensor categories and the modular closure
- Müger
(Show Context)
Citation Context ...is is useful in applications where this finite dimensionality is not known a priori, like in quantum field theory. Conversely, every ∗-category which is semisimple in our sense is a W ∗-category, cf. =-=[25]-=-. In a W ∗-category, every morphism s : X → Y has a polar decomposition s = pu, where p is positive and u a partial isometry. As a consequence, whenever Mor(X, Y ) contains a split monic (or isomorphi... |

25 | Locally compact quantum groups in the universal setting
- Kustermans
(Show Context)
Citation Context ...ns of aqg and lcqg. The latter results rely on the theory of infinite dimensional representations and corepresentations and of the construction of the universal corepresentation for lcqg developed in =-=[19]-=-. But none of this work touched on Tannaka-Krein reconstruction, since conjugates do not exist in the infinite dimensional case. Another type of reconstruction result for tensor C*-categories involvin... |

24 | Spherical categories
- Barrett, Westbury
- 1999
(Show Context)
Citation Context ...sumes that the category is equipped with an additional piece of structure, an ‘ε-structure’. A very similar notion naturally arises in recent axiomatizations of tensor categories with two-sided duals =-=[2]-=-, but it is quite superfluous if one works with ∗-categories. This will become clear in our treatment. (Yamagami has recently proved a result [44, Theorem 3.6] implying that, passing if necessary to a... |

23 |
Daele, C ∗ -algebraic quantum groups arising from algebraic quantum groups
- Kustermans, Van
- 1997
(Show Context)
Citation Context ... of a positive and leftinvariant functional, the Haar functional. All compact and all discrete quantum groups are aqg, and every aqg has a unique analytic extension to a locally compact quantum group =-=[20]-=-, having an equivalent von Neumann algebraic version [22]. A discrete multiplier Hopf ∗-algebra [35] can be shown to have a Haar functional rendering it a discrete aqg. The purpose of the present pape... |

22 | Co-amenability of compact quantum groups
- Bedos, Murphy, et al.
(Show Context)
Citation Context ...orep(Ar, ∆r) of unitary corepresentations on Hilbert spaces consists of Hilbert spaces K and unitaries V ∈ M(Ar ⊗B0(K)) such that (∆r ⊗ι)V = V13V23. For the exact correspondence thus established, see =-=[5]-=-. We will look at another way of obtaining a tensor ∗-category Rep f(A o s , ˆ ∆) from a compact aqg (A, ∆) which is equivalent to Corep f(A, ∆). This can sometimes come in very handy, especially when... |

22 |
On Hopf algebras and rigid monoidal categories
- Ulbrich
- 1990
(Show Context)
Citation Context ...or as Tannaka had effectively done [33] before the advent of category theory. The idea of considering the natural transformations of the embedding functor was generalized in the work of K.-H. Ulbrich =-=[34]-=-, where a given concrete tensor category is identified as the category of comodules over a Hopf algebra, cf. also [45]. 1While the role of the natural transformations of the embedding functor is obsc... |

22 |
Woronowicz: Compact matrix pseudogroups
- L
- 1987
(Show Context)
Citation Context ...roup is both abelian and compact. Pontryagin’s theorem can be stated more generally for locally compact quantum groups [21], a notion evolving out of Kac algebras [11] and compact matrix pseudogroups =-=[40]-=-. The theory of representations is most naturally developed in the language of tensor categories [24]. The category of finite dimensional representations of a compact group is a symmetric tensor ∗-cat... |

19 | Daele, Discrete quantum groups - Van - 1996 |

18 |
Lane: Categories for the Working Mathematician, 2nd edition
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- 1998
(Show Context)
Citation Context ...ct quantum groups [21], a notion evolving out of Kac algebras [11] and compact matrix pseudogroups [40]. The theory of representations is most naturally developed in the language of tensor categories =-=[24]-=-. The category of finite dimensional representations of a compact group is a symmetric tensor ∗-category with conjugates [9], and the Tannaka-Krein theorem tells us how to reconstruct the group from t... |

15 |
Daele: The Haar measure on some locally compact quantum groups
- Van
(Show Context)
Citation Context ...neracy. Furthermore, there exists a unique complex number µ such that ϕS 2 = µϕ. It can be proved that |µ| = 1, but it has recently been discovered that µ ̸= 1 for the quantum group version of ax + b =-=[38]-=-. Every aqg (A, ∆) has the property that to any a ∈ A, there exists c ∈ A such that ac = ca = a. Two aqg are said to be isomorphic if they are isomorphic as multiplier Hopf ∗-algebras and we use the s... |

14 |
Tannakian categories
- Breen
- 1994
(Show Context)
Citation Context ..., P. Podlés and Woronowicz, in defining a discrete quantum group, took the first step in this direction [28].) The approach to Tannakian categories motivated by algebraic geometry is well reviewed in =-=[6]-=-, where, however, only symmetric tensor categories are considered. Our approach to generalized Tannaka-Krein theory adopts the philosophy on quantum groups in [41, 43], meaning a self-dual category wi... |

11 |
Woronowicz: Tannaka–Krein duality for compact matrix pseudogroups
- L
- 1988
(Show Context)
Citation Context ...ct the group from the latter. In 1988, starting from a tensor ∗-category [12] with conjugates and admitting a generator, but assuming neither a symmetry nor a braiding, S. L. Woronowicz reconstructed =-=[41]-=- a compact matrix pseudogroup [40] having the given category as its category of finite dimensional unitary corepresentations. No general definition of a compact quantum group existed at the time. Once... |

10 |
An algebraic duality theory for multiplicative unitaries
- Doplicher, Pinzari, et al.
- 2001
(Show Context)
Citation Context ...reconstruction, since conjugates do not exist in the infinite dimensional case. Another type of reconstruction result for tensor C*-categories involving infinite dimensional objects was undertaken in =-=[10]-=- from the point of view of multiplicative unitaries and the regular corepresentation. For a discrete aqg (A, ∆) we establish a bijection between braidings of the category Repf(A, ∆) and Rmatrices in t... |

8 | Amenability and coamenability of algebraic quantum groups
- Bedos, Murphy, et al.
(Show Context)
Citation Context ...gebraic context and, while studying the amenability of quantum groups, the correspondence between tensor C*-categories of infinite dimensional representations and corepresentations was established in =-=[3, 4]-=-, for the various analytic versions of aqg and lcqg. The latter results rely on the theory of infinite dimensional representations and corepresentations and of the construction of the universal corepr... |

7 |
Rivano, Catégories Tannakiennes, Lecture notes in mathematics 265
- Saavedra
- 1972
(Show Context)
Citation Context ...t spaces. It is conceptually more satisfactory to start from an abstract tensor category together with a faithful tensor functor into the category of Hilbert spaces. In the work of N. Saavedra Rivano =-=[31]-=- the group associated with a concrete symmetric tensor category was identified as the group of natural monoidal automorphisms of the embedding functor as Tannaka had effectively done [33] before the a... |

7 |
On unitary representation theories of compact quantum groups
- Yamagami
- 1995
(Show Context)
Citation Context ...a Hopf algebra, cf. also [45]. 1While the role of the natural transformations of the embedding functor is obscure in Woronowicz’s approach, they appear at least implicitly in the work of S. Yamagami =-=[43]-=-, who considered representations of discrete quantum groups. His approach has two drawbacks. On the one hand, he assumes that the category is equipped with an additional piece of structure, an ‘ε-stru... |

6 |
Über den Dualitätssatz der nichtkommutativen topologischen Gruppen
- Tannaka
- 1939
(Show Context)
Citation Context ...vedra Rivano [31] the group associated with a concrete symmetric tensor category was identified as the group of natural monoidal automorphisms of the embedding functor as Tannaka had effectively done =-=[33]-=- before the advent of category theory. The idea of considering the natural transformations of the embedding functor was generalized in the work of K.-H. Ulbrich [34], where a given concrete tensor cat... |

5 |
On the equality of q-dimension and intrinsic dimension
- Roberts, Tuset
(Show Context)
Citation Context ...epresentations of (Uq(g), ˆ ∆) on finite dimensional Hilbert spaces is equivalent to Corepf (Aq, ∆). Also ˆ f ∈ Uq(g), so the intrinsic dimension can be read off conveniently for such representations =-=[30]-=-. We return to this issue in the next subsection. ✷ We now recast our results in the language of modules and comodules. Proposition 3.18 Any non-degenerate ∗-representation π of a discrete aqg (A, ∆) ... |

4 | Woronowicz: Compact quantum groups - L - 1998 |

2 | Frobenius duality in C ∗ -tensor categories - Yamagami - 2004 |

1 | Tuset: Representations of direct products of matrix algebras
- Guido, L
- 2001
(Show Context)
Citation Context ...) is a finite dimensional matrix algebra for every X ∈ C and Proposition 5.4 shows that NatE is a direct product of matrix algebras. The representation theory of such algebras is quite intricate, cf. =-=[13]-=-. There it is shown, among other results, that every irreducible representation of B = ∏ i∈I B(Hi) is equivalent to a projection pi iff the index set I has less than measurable cardinality. In practic... |

1 |
Examining the dual of an algebraic quantum group. funct-an/9704004
- Kustermans
(Show Context)
Citation Context ...e of Woronowicz and Yamagami on one hand and the purely algebraic ones on the other [34, 45]. In particular, making use of the universal unitary corepresentation of an aqg introduced by J. Kustermans =-=[18]-=-, we prove that the tensor ∗-category of ∗-representations (or modules) of a discrete aqg (A, ∆) is equivalent to the tensor ∗-category of unitary corepresentations (or comodules) of the dual compact ... |

1 |
Daele & Y. Zhang: Corepresentation theory of multiplier Hopf algebra I
- Van
- 1999
(Show Context)
Citation Context ...ional representations and corepresentations was already established in [28] and [18], but only for the objects of the respective categories, morphisms and the tensor structure were not considered. In =-=[39]-=- tensor structure and braiding were taken into account in a purely algebraic context and, while studying the amenability of quantum groups, the correspondence between tensor C*-categories of infinite ... |