Compact quantum groups

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Ergodic Actions of Universal Quantum Groups on Operator Algebras

We construct ergodic actions of compact quantum groups on C∗-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact...

2 5 M ar 1 99 8 Notes on Compact Quantum Groups

Compact quantum groups have been studied by several authors and from different points of view. The difference lies mainly in the choice of the axioms. In the end, the main results turn out to be the same. Nevertheless, the starting point has a strong influence on how the main results are obtained and on showing that certain examples satisfy these axioms. In these notes, we follow the approach o...

Fusion Rules for Representations of Compact Quantum Groups

The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q > 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of “continuous functions on it”, which is a Hopf C-algebra. A system of axioms for Hopf C-a...

Simple Compact Quantum Groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q ∈ GL(n,C) satisfying QQ̄ = ±In, n ≥ 2; (b) The quantum automorphism groups Aaut(B, τ) of finite dimensional C ∗-algebras B endowed with the canonical trace τ w...