Remarks on Hopf Images and Quantum Permutation Groups

Abstract

Abstract. Motivated by a question of A. Skalski and P.M. Sołtan ( ) 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 ( ) and study some group-theoretic properties of the quantum permutation group on points. is enables us not only to answer the aforementioned question in positive in case n = , k = , but also to classify the automorphisms of S + , describe all the embeddings O − ( ) ⊂ S + and show that all the copies of O − ( ) inside S + 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.

Quantum Increasing Sequences Generate Quantum Permutation Groups

“…We will proceed inductively. Case n = 4, k = 2 was solved in [13], let us assume n ≥ 5 and assume I + k,n−1 = S + n−1 for all 2 ≤ k ≤ n − 3. Firstly, S n ⊆ I + k,n from Lemma 2.6.…”

“…In this manuscript we use those result to show that I + k,n = S + n for all n ≥ 4 and 2 ≤ k ≤ n − 2, answering a question of Skalski and So ltan from [18]. To do so, we extend the criterion we gave in [12] to cover a wider class of subsets of the generating sets. Informally, we show that I k,n ⊂ I + k,n and that…”

“…In general, [18,Question 7.3] asks for the complete description of all I + k,n and emphasises the case n = 4 and k = 2 as the first non-trivial case to study. We gave a positive answer in this case in [12] and explained that in general S n ⊂ I + k,n ⊂ S + n , the left inclusion being strict whenever 2 ≤ k ≤ n − 2. Banica's conjecture asks whether there exists a compact quantum group G satisfying S n ⊂ G ⊂ S + n such that both inclusions are strict: conjecturally only one of them should be.…”

Thoma type results for discrete quantum groups

“…These subgroups, which are all coamenable, were fully classified in [3], and most of them can be investigated by using the above results. The examples which are not covered yet by our results consist in certain finite quantum groups, appearing as cocycle twists [19], plus O −1 2 , SO −1 3 , which can be probably investigated by using the fibers of the Pauli matrix representation [3], [6], [18], [22].…”