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 B(u) (Q) for Q is an element of GL(n,C) satisfying Q (Q) over bar = +/- l(n), n >= 2; (b) The quantum automorphism groups A(aut)(B, tau) of finite-dimensional C*-algebras B endowed with the canonical trace tau when dim(B) >= 4, including the quantum permutation groups A(aut)(X(n)) on n points (n >= 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). (c) 2008 Elsevier Inc. All rights reserved.