Simple Compact Quantum Groups I

Abstract

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 $B_u(Q)$ for $Q\in GL(n,\mathbb{C})$ satisfying $Q\bar{Q}=\pm I_n$, $n\ge 2$; (b) The quantum automorphism groups $A_{aut}(B,\tau )$ of finite dimensional $C^{\ast }$-algebras $B$ endowed with the canonical trace $\tau$ %endowed with a tracial functional $tr$ when $\dim (B)\ge 4$, including the quantum permutation groups $A_{aut}(X_n)$ on $n$ points ($n\ge 4$); (c) The standard deformations $K_q$ of simple compact Lie groups $K$ and their twists $K_q^u$, as well as Rieffel's deformation $K_J$.

AMS-LATEX file, 49 pages

References in corpus (1)

• The hyperoctahedral quantum group

Cited by in corpus (13)

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• Free unitary groups are (almost) simple
• The Ergodic Theorem for Random Walks on Finite Quantum Groups
• Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map
• Deformations of spectral triples and their quantum isometry groups via monoidal equivalences
• Connectedness and Gaussian Parts for Compact Quantum Groups
• On Square Roots of the Haar State on Compact Quantum Groups
• Connectedness and irreducibility of compact quantum groups
• On the problem of classifying simple compact quantum groups