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QUANTUM INCREASING SEQUENCES GENERATE QUANTUM PERMUTATION GROUPS

Part of: Groups and algebras in quantum theory Linear algebraic groups and related topics Selfadjoint operator algebras Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  27 September 2019

PAWEŁ JÓZIAK Open the ORCID record for PAWEŁ JÓZIAK [Opens in a new window]
PAWEŁ JÓZIAK*
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland e-mail: p.joziak@mini.pw.edu.pl
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Abstract

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We answer a question of Skalski and Sołan (2016) about inner faithfulness of the Curran’s map of extending a quantum increasing sequence to a quantum permutation. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran’s map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon (2018).

MSC classification

Primary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 20G42: Quantum groups (quantized function algebras) and their representations
Secondary: 81R50: Quantum groups and related algebraic methods 16T20: Ring-theoretic aspects of quantum groups

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 631 - 639
Copyright
© Glasgow Mathematical Journal Trust 2019

References

Banica, T., Truncation and duality results for Hopf image algebras, Bull. Pol. Acad. Sci. Math., 62 (2014), 161–180.CrossRefGoogle Scholar
Banica, T., Homogeneous quantum groups and their easiness level, arXiv e-prints. arXiv:1806.06368 (2018).Google Scholar
Banica, T. and Bichon, J., Quantum groups acting on 4 points, J. Reine Angew. Math., 626 (2009), 75–114.Google Scholar
Banica, T. and Bichon, J., Hopf images and inner faithful representations, Glasg. Math. J., 52 (2010), 677–703.CrossRefGoogle Scholar
Banica, T. and Speicher, R., Liberation of orthogonal Lie groups, Adv. Math., 222 (2009), 1461–1501.CrossRefGoogle Scholar
Bichon, J. and Yuncken, R., Quantum subgroups of the compact quantum group SU-1(3), Bull. Lond. Math. Soc., 46 (2014), 315–328.CrossRefGoogle Scholar
Brannan, M., Chirvasitu, A., and Freslon, A., Topological generation and matrix models for quantum reflection groups, arXiv e-prints, arXiv:1808.08611 (2018).Google Scholar
Brannan, M., Collins, B., and Vergnioux, R., The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc., 369 (2017), 3799–3819.CrossRefGoogle Scholar
Chirvasitu, A., Residually finite quantum group algebras, J. Funct. Anal., 268 (2015), 3508–3533.CrossRefGoogle Scholar
Curran, S., A characterization of freeness by invariance under quantum spreading, J. Reine Angew. Math., 659 (2011), 43–65.Google Scholar
Daws, M., Kasprzak, P., Skalski, A., and Sołtan, P. M., Closed quantum subgroups of locally compact quantum groups, Adv. Math., 231 (2012), 3473–3501.CrossRefGoogle Scholar
Józiak, P., Hopf images in Locally Compact Quantum Groups, PhD Thesis (IMPAN, 2017).CrossRefGoogle Scholar
Józiak, P., Remarks on Hopf images and quantum permutation groups , Canad. Math. Bull., 61 (2018), 301–317.CrossRefGoogle Scholar
Józiak, P., Kasprzak, P., and Sołtan, P. M., Hopf images in locally compact quantum groups, J. Math. Anal. Appl., 455 (2017), 141–166.CrossRefGoogle Scholar
Köstler, C. and Speicher, R., A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys., 291 (2009), 473–490.CrossRefGoogle Scholar
Kyed, D. and Sołtan, P. M., Property (T) and exotic quantum group norms, J. Noncommut. Geom., 6 (2012), 773–800.CrossRefGoogle Scholar
Neshveyev, S. and Tuset, L., Compact quantum groups and their representation categories, vol. 20. Cours Spécialisés [Specialized Courses] (Société Mathématique de France, Paris, 2013).Google Scholar
Podleś, P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum and groups, Comm. Math. Phys., 170 (1995), 1–20.CrossRefGoogle Scholar
Skalski, A. and Sołtan, P. M., Quantum families of invertible maps and related problems, Canad. J. Math., 68 (2016), 698–720.CrossRefGoogle Scholar
Timmermann, T., An Invitation to Quantum Groups and Duality (EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008). From Hopf algebras to multiplicative unitaries and beyond.CrossRefGoogle Scholar
Wang, S., Quantum symmetry groups of finite spaces, Comm. Math. Phys., 195 (1998), 195–211.CrossRefGoogle Scholar
Woronowicz, S. L., Compact quantum groups, in Symétries quantiques (Houches, L., Editor) (North-Holland, Amsterdam, 1998), 845–884.Google Scholar