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

Józiak, Paweł

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 301-317

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-028-2

Mots-clés : 20G42, 81R50, 46L89, 16W35, Hopf image, quantum permutation group, compact quantum group

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