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# Remarks on Hopf Images and Quantum Permutation Groups $S_n^+$

Published:2017-06-07
Printed: Jun 2018
• Paweł Józiak,
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00--656 Warszawa, Poland
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## Abstract

Motivated by a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the 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 $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, 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: Hopf image, quantum permutation group, compact quantum group
 MSC Classifications: 20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 81R50 - Quantum groups and related algebraic methods [See also 16T20, 17B37] 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22] 16W35 - Ring-theoretic aspects of quantum groups (See also 17B37, 20G42, 81R50)

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