Regular Objects, Multiplicative Unitaries and Conjugation

Pinzari, Claudia; Roberts, John Η.

doi:10.1142/s0129167x02001423

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…where R A is the unitary antipode for A (see [20]). As remarked already in the previous section, note that we can take the square θ 1/2 X of the positive diagonalizable operator θ X , i.e.…”

Section: Proposition 224 the Natural Transformation N ⊗ Is Unitarymentioning

confidence: 99%

Galois objects for algebraic quantum groups

Commer

2009

Journal of Algebra

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The basic elements of Galois theory for algebraic quantum groups were given in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by Van Daele and Zhang. In this paper, we supplement their results in the special case of Galois objects: algebras equipped with a Galois coaction by an algebraic quantum group, such that only the scalars are coinvariants. We show how the structure of these objects is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We show how to reflect the quantum group across a Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.

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Section: Proposition 224 the Natural Transformation N ⊗ Is Unitarymentioning

confidence: 99%

Galois objects for algebraic quantum groups

Commer

2009

Journal of Algebra

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“…Remark For a discussion of antilinear arrows see [19], where it is pointed out that J σ ⊗ J ρ • θ ρ,σ is a natural tensor product to use for antilinear arrows. Proof.…”

Section: Theoremmentioning

confidence: 99%

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C *-Algebras

Pinzari

Roberts

2007

Commun. Math. Phys.

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The spectral functor of an ergodic action of a compact quantum group G on a unital C * -algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor * -functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C * -algebra.As an application, we associate an ergodic G-action on a unital C *algebra to an inclusion of Rep(G) into an abstract tensor C * -category T.If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and T has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G.If a tensor C * -category has a Hecke symmetry making an object ρ of dimension d and µ-determinant 1 then there is an ergodic action of SµU (d) on a unital C * -algebra having the (ι, ρ r ) as its spectral subspaces. The special case of SµU (2) is discussed.

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“…We call such an object a natural right absorber, avoiding the overused adjective "regular". Going beyond [8], we show that different natural right absorbers give isomorphic multiplicative unitaries with respect to the morphisms of C * -quantum groups defined in [3,7]. We also add a further equivalent description of such quantum group morphisms through functors between representation categories, and we show that isomorphic multiplicative unitaries generate isomorphic C * -quantum groups.…”

Section: Natural Right Absorbers In Hilbert Space Tensor Categoriesmentioning

confidence: 91%

“…We are going to recall the notion of a (right) regular object of a tensor category from [8]. We call such an object a natural right absorber, avoiding the overused adjective "regular".…”

Section: Natural Right Absorbers In Hilbert Space Tensor Categoriesmentioning

confidence: 99%

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Braided multiplicative unitaries as regular objects

Meyer¹,

Roy²

Advanced Studies in Pure Mathematics

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We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic C * -quantum groups. In particular, a C * -quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of C * -quantum groups.2000 Mathematics Subject Classification. 46L89 (81R50 18D10). Key words and phrases. quantum group, braided quantum group, multiplicative unitary, braided multiplicative unitary, tensor category, quantum group representation, quantum group morphism, Tannaka-Krein Theorem.

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Regular Objects, Multiplicative Unitaries and Conjugation

Cited by 4 publications

References 17 publications

Galois objects for algebraic quantum groups

Galois objects for algebraic quantum groups

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C *-Algebras

Braided multiplicative unitaries as regular objects

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