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Remarks on Hopf Images and Quantum Permutation Groups$S_{n}^{+}$

Published online by Cambridge University Press:  20 November 2018

Paweł Józiak
Paweł Józiak*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, 00-656 Warszawa, Poland and Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland, e-mail: pjoziak@impan.pl
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Abstract

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Motivated by a question of A. Skalski and P. M. Sołtan (2016) about inner faithfulness of S. Curran’s map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on points. This enables us not only to answer the aforementioned question in the positive for the case where $n\,=\,4,\,k\,=\,2$, but also to classify the automorphisms of $S_{4}^{+}$, describe all the embeddings ${{O}_{-1}}(2)\,\subset \,S_{4}^{+}$ and show that all the copies of ${{O}_{-1}}(2)$ inside $S_{4}^{+}$are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

Keywords

20G4281R5046L8916W35Hopf imagequantum permutation groupcompact quantum group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 2 , 01 June 2018 , pp. 301 - 317
Copyright
Copyright © Canadian Mathematical Society 2018

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