# finite quantum group

###### Definition 0.1.

A finite quantum group $Q_{Gf}$ is a pair $(\mathbb{H},\Phi)$ of a finite-dimensional $C^{*}$-algebra $\mathbb{H}$ with a comultiplication $\Phi$ such that $(\mathbb{H},\Phi)$ is a Hopf ${}^{*}$-algebra.

Note that one obtains the dual Hopf algebra of a commutative, finite quantum group via Fourier transformation of the group’s elements.

## References

• 1 Abe, E., Hopf Algebras, Cambridge University Press, 1977.
• 2 Sweedler, M. E., Hopf Algebras, W.A. Benjamin, inc., New York, 1969.
• 3 Kustermans, J., Van Daele, A., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Int. J. of Math. 8 (1997), 1067-1139.
• 4 Lance, E.C., An explicit description of the fundamental unitary for $SU(2)_{q}$, Commun. Math. Phys. 164 (1994), 1-15.
 Title finite quantum group Canonical name FiniteQuantumGroup Date of creation 2013-03-22 18:24:10 Last modified on 2013-03-22 18:24:10 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 17 Author bci1 (20947) Entry type Definition Classification msc 46L05 Classification msc 81R15 Classification msc 81R50 Synonym quantum group Synonym dual of a finite Hopf algebra Related topic CompactQuantumGroup Related topic HopfAlgebra Related topic GrassmanHopfAlgebrasAndTheirDualCoAlgebras Related topic CompactMatrixQuantumGroup Defines comultiplication in a quantum group Defines dual of a finite Hopf algebra