Quantum group-twisted tensor products of C*-algebras. II

Meyer, Ralf; Roy, Sutanu; Woronowicz, S. L.

doi:10.4171/jncg/250

Cited by 16 publications

(16 citation statements)

References 21 publications

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“…In the present paper we show that any monoidal structure on C * G with these properties is related to a unitary R-matrix in the way described in [7]. What surprises in this result is the fact that it could be obtained in so general and abstract setting.…”

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confidence: 53%

“…Natural properties of a monoidal structure on C * G Let G = (A, ∆) be a quasitriangular locally compact quantum group. In [7] we introduced a monoidal structure ⊠ on the category C * G . It has the following properties:…”

Section: Monoidal Structuresmentioning

confidence: 99%

“…The monoidal structure constructed in the latter paper have three very natural properties: monoidal product is a crossed product (Property 1), monoidal product of injective morphisms is injective (Property 2) and monoidal product reduces to the minimal tensor product when one of the involved C * -algebras is equipped with a trivial action of the group (Property 3). See Proposition 4.6 in [7].…”

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confidence: 99%

“…Monoidal structures considered in [14,10,8,7] have some common very interesting properties. In section 6 we list these properties and investigate monoidal structures with these properties.…”

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Monoidal category of C*-algebras

Woronowicz¹

2016

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We consider the category of C * -algebras equipped with actions of a locally compact quantum group. We show that this category admits a monoidal structure satisfying certain natural conditions if and only if the group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices. To prove this result we use only very natural properties imposed on considered monoidal structures. We assume that monoidal product is a crossed product, monoidal product of injective morphisms is injective and that monoidal product reduces to the minimal tensor product when one of the involved C * -algebras is equipped with a trivial action of the group. No a priori form of monoidal product is used. IntroductionOne of the paradigms of quantum theory says that the algebra of observables associated with a composed system is the tensor product of algebras associated with the parts of the system. This is the simplest monoidal functor.Given C * -algebras X and Y one may consider the C * -algebra X ⊗ Y . If X and Y are equipped with actions of a locally compact group G, then there exists unique action of G on X ⊗ Y such that natural embeddings of X and Y into X ⊗ Y intertwine the actions of G. In the categorical language: tensor product ⊗ defines a monoidal structure on the category C * G of all C * -algebras equipped with the action of G. This is no longer the case if G is a quantum group. Let G be a locally compact quantum group. We shall show that the category of C * G admits a monoidal structure satisfying certain natural conditions if and only if the group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices. In general the monoidal structure ⊠ does not coincide with ⊗.Early examples of monoidal structures on C * G are given in [14] (for G = Z × S 1 ) and [10] (for G = R). To construct monoidal structure on C * G was not a trivial task. Let R be a unitary R-matrix. To define monoidal productMethods used in [10,14] took into account particular properties of the considered groups and gave no indication how to proceed in general case. A decisive step was made by Ryszard Nest and Christian Voigt in [8]. They showed that the intelligent use of Podleś condition (continuity of action) solves the problem. Nest and Voigt worked with locally compact quantum groups dual to Drinfeld doubles. The case of any quasitriangular locally compact quantum group was Date: June 2016. 2000 Mathematics Subject Classification. 46L55 (81R50). Key words and phrases. C * -algebra, Action of quantum groups, Crossed product, Monoidal structure, Unitary R-matrix.

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confidence: 53%

Section: Monoidal Structuresmentioning

confidence: 99%

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Monoidal category of C*-algebras

Woronowicz¹

2016

Preprint

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“…Remarkably, we get a candidate for bicharacter W of G and the dual braided quantum group. The attempt of formulating an analytic theory of braided quantum groups was undertaken in [12] and continued in [10]. For the algebraic counterpart of the notion of a braided quantum group see [7,Section 9].…”

Section: Introductionmentioning

confidence: 99%

Rieffel Deformation of Tensor Functor and Braided Quantum Groups

Kasprzak

2015

Commun. Math. Phys.

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We apply Rieffel deformation to C * -tensor product viewed as a functor on the category of C * -algebras with an abelian group action. In the case of the Rieffel deformation of a quantum group with the action by automorphisms the deformed tensor product enables us to view the deformed object as a braided quantum group. We construct a bicharacter for a braided quantum group and the dual braided quantum group. We employ our method to get a braided quantum Minkowski space. Its description in terms of the deformed space-time coordinates is provided.

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