Remarks on Hopf Images and Quantum Permutation Groups

Abstract

AbstractMotivated 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 = z, but also to classify the automorphisms of , describe all the embeddings O−1(2) ⊂ and show that all the copies of O−1(2) ⊂ 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.