Homogeneous quantum groups and their easiness level

Topological generation and matrix models for quantum reflection groups

“…(Case 2: N = 5) According to [5,Theorem 7.10] the inclusion S 5 < S + 5 admits no intermediate quantum groups. Since S + 4 < S + 5 is not a quantum subgroup of S 5 , we indeed have S + 5 = S 5 , S + 4 .…”

“…This difference turns out to be a fundamental obstruction to a straightforward extension of the inductive arguments of [24,22]. To bypass this issue, we make essential use of a recent remarkable result of Banica [5,Theorem 7.10] which establishes that there is no intermediate quantum subgroup for the inclusion S 5 < S + 5 . The maximality of the inclusion S N < S + N is widely conjectured to be true for all N , and the case N = 5 solved by Banica in [5] represents a major advancement on this conjecture.…”

“…To bypass this issue, we make essential use of a recent remarkable result of Banica [5,Theorem 7.10] which establishes that there is no intermediate quantum subgroup for the inclusion S 5 < S + 5 . The maximality of the inclusion S N < S + N is widely conjectured to be true for all N , and the case N = 5 solved by Banica in [5] represents a major advancement on this conjecture. It is also interesting to note that Banica's proof of the maximality of the inclusion S 5 < S + 5 is based on a reduction of this problem to the seemingly different problem of classifying II 1 -subfactors at index 5.…”

“…It is an artifact of our proof that we are unable to settle the cases N ∈ [6,9] in the above theorems. Our arguments rely heavily on our topological generation results for S + N together with the crucial result of Banica [5] stating that the inclusion S 5 < S + 5 is maximal. The remainder of the paper is organized as follows.…”

Quantum groups under very strong axioms

“…Alond the same lines, using some even weaker conditions, of type A N ⊂ G, make sense as well, at least theoretically. For some comments here, we refer to [4]. This is something philosophical.…”