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 QQ¯ =±In, n 2; (b) The quantum automorphism groups Aaut(B, τ ) of finite-dimensional C∗-algebras B endowed with the canonical trace τ when dim(B) 4, including the quantum permutation groups Aaut(Xn) on n points (n 4); (c) The standard deformations Kq of simple compact Lie groups K and their twists Ku q , as well as Rieffel’s deformation KJ . © 2008 Elsevier Inc. All rights reserved