Simple Compact Quantum Groups I

Details Simple Compact Quantum Groups I Simple Compact Quantum Groups I

2008-10-31

arxiv.org/abs/0810.5734v1

J. Funct. Anal. 256 (2009), no. 10, 3313--3341

Mathematics

Quantum Algebra

AMS-LATEX file, 49 pages

Scientific paper

10.1016/j.jfa.2008.10.020

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

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