# compact quantum group

###### Definition 0.1.

A compact quantum group, $Q_{CG}$ is defined as a particular case of a locally compact quantum group $Q_{LCG}$, that is, a quadruple $(A,\Delta,\mu,\nu)$, where $A$ is either a $C^{*}$– or a $W^{*}$algebra equipped with a co-associative comultiplication (http://planetmath.org/WeakHopfCAlgebra2) $\Delta:A\to A\otimes A$, 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 $Q^{T}_{CG}$, 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