Remarks on Hopf Images and Quantum Permutation Groups $S_{n}^{+}$
Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 301-317
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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.
Mots-clés :
20G42, 81R50, 46L89, 16W35, Hopf image, quantum permutation group, compact quantum group
Józiak, Paweł. Remarks on Hopf Images and Quantum Permutation Groups $S_{n}^{+}$. Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 301-317. doi: 10.4153/CMB-2017-028-2
@article{10_4153_CMB_2017_028_2,
author = {J\'oziak, Pawe{\l}},
title = {Remarks on {Hopf} {Images} and {Quantum} {Permutation} {Groups} $S_{n}^{+}$},
journal = {Canadian mathematical bulletin},
pages = {301--317},
year = {2018},
volume = {61},
number = {2},
doi = {10.4153/CMB-2017-028-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-028-2/}
}
TY - JOUR
AU - Józiak, Paweł
TI - Remarks on Hopf Images and Quantum Permutation Groups $S_{n}^{+}$
JO - Canadian mathematical bulletin
PY - 2018
SP - 301
EP - 317
VL - 61
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-028-2/
DO - 10.4153/CMB-2017-028-2
ID - 10_4153_CMB_2017_028_2
ER -
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