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