# compact quantum group

###### Definition 0.1.

A *compact quantum group, ${Q}_{C\mathit{}G}$* is defined as a particular case of a locally compact quantum group^{} ${Q}_{LCG}$, that is, a quadruple $(A,\mathrm{\Delta},\mu ,\nu )$, where $A$ is either a ${C}^{*}$– or a
${W}^{*}$– algebra^{} equipped with a co-associative comultiplication (http://planetmath.org/WeakHopfCAlgebra2)
$\mathrm{\Delta}:A\to A\otimes A$, and two faithful^{} semi-finite normal weights, $\mu $ and $\nu $ –right and -left Haar measures, and also when the object space $\mathbf{O}$ of the latter ${Q}_{LCG}$ is replaced by a compact topological space ${Q}_{CG}^{T}$, instead of being a locally compact topological space like ${Q}_{LCG}$.

## References

- 1 A. Maes, and A. VanDaele. 1998. http://arxiv.org/PS_cache/math/pdf/9803/9803122v1.pdfNotes on Compact Quantum Groups., $arxiv.org.math-FA-9803122v1$, 43 pp.

Title | compact quantum group |

Canonical name | CompactQuantumGroup |

Date of creation | 2013-03-22 18:24:07 |

Last modified on | 2013-03-22 18:24:07 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 25 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 81R15 |

Classification | msc 81R50 |

Classification | msc 46L05 |

Synonym | quantum group^{} |

Synonym | compact matrix quantum group |

Related topic | CAlgebra3 |

Related topic | QuantumOperatorAlgebrasInQuantumFieldTheories |

Related topic | FiniteQuantumGroup |

Related topic | DualityInMathematics |

Related topic | LocallyCompactQuantumGroup |

Related topic | QuantumGroups |

Related topic | GelfandTransform |