Examples of non-compact quantum group actions

Sołtan, Piotr M.

doi:10.1016/j.jmaa.2010.06.045

Cited by 19 publications

(9 citation statements)

References 21 publications

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“…We will be also interested in its topological version, which should be a C * -algebra contained in L ∞ (G H), on which G naturally acts (for the notion of action of a locally compact quantum group on a C * -algebra we refer for example to [32] or [35]). In general the problem of its existence remains open, but for regular quantum groups it was solved in [35], where the following result was shown.…”

Section: All the Notions And Statements Above Have Natural Counterpar...mentioning

confidence: 99%

Open quantum subgroups of locally compact quantum groups

Kalantar

Kasprzak

Skalski

2016

Advances in Mathematics

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Abstract. The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C * -algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence with normal compact quantum subgroups of the dual quantum group.The theory of locally compact quantum groups, formulated in the language of operator algebras, is a rapidly developing field closely related to abstract harmonic analysis, and with various connections to noncommutative geometry, quantum probability and other areas of 'noncommutative' mathematics. A locally compact quantum group G is a virtual object, studied via its 'algebras of functions': a C * -algebra C 0 (G), playing the role of the algebra of continuous functions on G vanishing at infinity, and a von Neumann algebra L ∞ (G), viewed as the algebra of essentially bounded measurable functions on G; both of these are equipped with coproducts, operations encoding the 'multiplication operation' of G. It is often essential to study both of the avatars of G mentioned above (as well as the universal counterpart of C 0 (G), the C * -algebra C u 0 (G)) at the same time (see for example [15], [35]). This will also be the case in this article.In recent years we have seen an increased interest in the notion of morphisms between locally compact quantum groups H and G. These also have various incarnations (see [27]), one of which is given by C * -algebra morphisms π ∈ Mor(C u 0 (G), C u 0 (H)) intertwining the respective coproducts. It is then natural to consider what it means that a given morphism between H and G has closed image and moreover is a homeomorphism onto this imagein other words, identifies H with a closed (quantum) subgroup of G. This problem was studied in depth in the article [6], where two alternative definitions, due respectively to Vaes and Woronowicz, formulated respectively in the von Neumann algebraic and C * -algebraic language, were compared and interpreted in several special cases. Among the main results of [6] was the proof that a closed quantum subgroup in the sense of Vaes is always a closed quantum subgroup in the sense of Woronowicz, and in fact in many cases (classical, dual to classical, compact, discrete) the two definitions coincide. Understanding the concept of a closed quantum subgroup forms a very natural step in the development of the theory and was key in building and applying the induction theory for representations of quantum groups (as formulated in [22], [35]).In the current work, which can be naturally viewed as a continuation of H of a locally compact group G is automatically closed, a fact that leads to simplification of many questions concerning relations between representation theoretical and harmonic analytical properties of H and G. Perhaps, the main reason for the latter is that openness of H in G ...

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Section: All the Notions And Statements Above Have Natural Counterpar...mentioning

confidence: 99%

Open quantum subgroups of locally compact quantum groups

Kalantar

Kasprzak

Skalski

2016

Advances in Mathematics

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“…• This is also the case when H is a compact. This is essentially proved in [28,Theorem 5.1]. See also the remark after Theorem 4.7.…”

Section: 2mentioning

confidence: 80%

“…In the classical case, a homogeneous space G/H can be also constructed by taking the quotient space of G with the right H-action, or equivalently by taking the fixed point algebra of C 0 (G) with the right H-coaction. When the quantum subgroup H is compact, this method still works as shown in [28]. However, in the general non-compact case, we have technical subtleties to construct C r 0 (G/H) as a fixed point algebra.…”

Section: Introductionmentioning

confidence: 99%

Induced coactions along a homomorphism of locally compact quantum groups

Kitamura¹

2021

Preprint

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We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group. Contents 1. Introduction 1 2. Preliminaries 3 3. Formal aspects of induced coactions 6 4. Induced coactions along a proper homomorphism 10 5. Analogue of base change 16 6. Morita and K-theoretic properties 21 7. Example: double crossed products with a discrete abelian group 25 Appendix A. Constructions for locally compact quantum groups and coactions 28 References 32

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“…The proof of Statement (2) follows the lines of analogous statements for non-braided quantum groups (see e.g. [20,Section 5]). Since braided products (in the sense of Section 2) are not as familiar as ordinary tensor products, we will spell out all the details.…”

Section: The Quotient Spherementioning

confidence: 88%

Podleś spheres for the braided quantum SU(2)

Sołtan

2020

Linear Algebra and its Applications

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Examples of non-compact quantum group actions

Cited by 19 publications

References 21 publications

Open quantum subgroups of locally compact quantum groups

Open quantum subgroups of locally compact quantum groups

Induced coactions along a homomorphism of locally compact quantum groups

Podleś spheres for the braided quantum SU(2)

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