Remark on Hopf images in quantum permutation groups S_n⁺

2016/11/28 by Józiak, Paweł
#20G42 Secondary: 16W35 #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 46L89 #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1611.09211

Abstract

Motivated by a question of A.~Skalski and P.M.~Sołtan about inner faithfulness of the S.~Curran's map, we revisit the results and techniques of T.~Banica and J.~Bichon's Crelle paper 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 S₄⁺, describe all the embeddings O_-1(2)⊂ S₄⁺ and show that all the copies of O_-1(2) inside S₄⁺ are conjugate. We then use these results to show that the criterion we applied to answer the aforementioned question does not admit converse.

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