Unitary Easy Quantum Groups: The Free Case and the Group Case

Abstract

Abstract. Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C * -algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group O + n . We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of U + n . In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups.We also study the notion of easy groups. IntroductionIn order to provide a new notion of symmetries adapted to the situation in operator algebras, Woronowicz introduced compact matrix quantum groups in 1987 in [Wor87]. The idea is roughly to take a compact Lie group G ⊆ M n (C) and to pass to the algebra C(G) of continuous functions over it. It turns out that this algebra fulfills all axioms of a C * -algebra with the special feature that the multiplication is commutative. If we now also dualize main properties of the group law µ : G×G → G to properties of a map ∆ : C(G) → C(G×G) ∼ = C(G)⊗C(G) (the comultiplication), and consider noncommutative C * -algebras equiped with such a map ∆, we are right in the heart of the definition of a compact matrix quantum group (see Section 3.1 for details). It generalizes the notion of a compact matrix group. The reader not familiar with quantum groups might also first jump to Section 5 and read it as a motivation.Date: December 2, 2015. 2010 Mathematics Subject Classification. 20G42 (Primary); 05A18, 46L54 (Secondary).

Weingarten integration over noncommutative homogeneous spaces

“…Thus the result follows from Proposition 8.1, and from the BercoviciPata bijection result for truncated characters for this latter liberation operation [6,16].…”

“…Let us recall from [6,16] Now since the sum on the right equals |J| |π 1 ∨...∨πs| , this gives the result.…”

“…Regarding easiness in general, we refer to [6,14,16]. In the context of the present paper, let us go back to the Schur-Weyl considerations in Section 4:…”

“…See [6,16]. Now back to our homogeneous space questions, we have: C,+ appears by imposing the relations…”

Models of quantum permutations

“…Orthogonal easy quantum groups have been defined for the first time in [BS09] and they have been generalized in [TW18] and [TW17] to unitary easy quantum groups. This section is aimed for readers familiar with easy quantum groups and we refer to the references above for more details.…”