The canonical central exact sequence for locally compact quantum groups

Kasprzak, Paweł; Skalski, Adam; Sołtan, Piotr M.

doi:10.1002/mana.201600208

Cited by 7 publications

(6 citation statements)

References 33 publications

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“…The topics of quantum subgroups are thoroughly covered in [Daw+12] and information on normal quantum subgroups, inner automorphisms etc. can be found in [KSS16] and references therein. We will at some point use the theory of locally compact quantum groups in the sense of Kustermans and Vaes ([KV00]).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

et al. 2018

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To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e. when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R. Vergnioux.

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Section: Notation and Preliminariesmentioning

confidence: 99%

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

et al. 2018

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“…see [17]. In what follows we shall describe the quantum analog of the quotient of G by its commutator subgroup.…”

Section: Lattice Of Closed Quantum Subgroups: Basic Factsmentioning

confidence: 99%

Fundamental isomorphism theorems for quantum groups

Chirvăsitu

Hoche

Kasprzak

2017

Expositiones Mathematicae

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Abstract. The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-Hölder theorem.We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa.The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact / discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.

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“…• There are notions of (closed [36,Definition 2.6]) normal [37,Definition 2.10] and central [20,Definition 2.3] quantum subgroups H ≤ G.…”

Section: Introductionmentioning

confidence: 99%

“…The last equality in (2-20) is nothing but (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), whereas the first equality will follow once we have (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21); it thus remains to prove the latter. For that purpose, note that for…”

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confidence: 97%

Quantum relative modular functions

Chirvăsitu¹

2022

Preprint

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Let H G be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with L ∞ (G) that, roughly, measures the failure of G to act measure-preservingly on H by conjugation. The triviality of that element is equivalent to the condition that G and G/H have the same modular element, by analogy with the classical situation. This condition is automatic if H ≤ G is central, and in general implies the unimodularity of H.We also describe a bijection between strictly positive group-like elements δ affiliated with C 0 (G) and quantum-group morphisms G → (R, +), with the closed image of the morphism easily described in terms of the spectrum of δ. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.

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The canonical central exact sequence for locally compact quantum groups

Cited by 7 publications

References 33 publications

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

Fundamental isomorphism theorems for quantum groups

Quantum relative modular functions

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