The combinatorics of an algebraic class of easy quantum groups

Abstract

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group Sn and the orthogonal group On as well as Wang's quantum permutation group S + n and his free orthogonal quantum group O + n . In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.

A duality principle for noncommutative cubes and spheres

“…g j l , ∀i, j, k, l, ker( i j ) ∈ D(k, l) As explained in [47], the correspondences Γ → D and D → Γ are bijective, and inverse to each other, at N = ∞. We have in fact the following result, from [46], [47], [48]: Proposition 1.8. We have correspondences between:…”

“…We refer to the body of the paper for the precise statements of the results. The proofs are based on our previous work on noncommutative spheres in [3], [4], [8], [16], and on the classification work of Raum and Weber in [46], [47], [48], [49]. Let us also mention that, at the axiomatic level, we use a formalism inspired from [23], [40], [45].…”

The classification of tensor categories of two-colored noncrossing partitions

“…In the unitary case, hyperoctahedral categories should be those containing •••• but not ↑ • ⊗ ↑ • . It is likely that the classification of non-hyperoctahedral categories is more or less immediately doable, but as for the hyperoctahedral ones, it is unclear whether the methods of [RW14] and [RW13] can be applied directly.…”

The Full Classification of Orthogonal Easy Quantum Groups

“…We say that a category C of partitions is hyperoctahedral, if the four block partition ⊓⊓⊓ is contained in C but the double singleton ↑ ⊗ ↑ is not. The non-hyperoctahedral case was treated in [BaCuSp10] and [We13], whereas the class of so called group-theoretical hyperoctahedral categories has been studied in [RaWe13a] and [RaWe13b]. For the remaining case, we obtain a complete classification in the present article using the following partitions π k given by k four blocks on 4k points: π k = a 1 .…”

“…Theorem 2 (Theorem 5.4). If G is an orthogonal easy quantum group, its corresponding category of partitions (i) either is non-hyperoctahedral (and hence it is one of the 13 cases of [We13]), (ii) or it is hyperoctahedral and contains ⊓ − ⊓ and hence is group-theoretical (see [RaWe13a], [RaWe13b]). (iii) or it coincides with ⟨π l , l ≤ k⟩ for some k ∈ {1, 2, .…”