Simple Compact Quantum Groups I

TL;DR

This paper introduces the concept of simple compact quantum groups and provides examples of such groups, including certain universal quantum groups, quantum automorphism groups, and deformations of classical Lie groups, expanding the understanding of quantum symmetries.

Contribution

The paper defines simple compact quantum groups and demonstrates their existence through various non-trivial examples, including universal quantum groups, quantum automorphism groups, and deformations of classical groups.

Findings

Universal quantum groups $B_u(Q)$ are simple for specific $Q$.

Quantum automorphism groups $A_{aut}(B, au)$ are simple for certain finite-dimensional $C^*$-algebras.

Deformations of classical Lie groups $K_q$, $K_q^u$, and $K_J$ are shown to be simple.

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 $Q \bar{Q} = \pm I_n$, $n \geq 2$; (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 $K_J$.