PlanetPhysics/Finite Quantum Group 2

Finite Quantum (Hopf) Algebra

Recall that: 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.

A finite quantum algebra $A_{Gf}$ is the dual of a finite quantum group $Q_{Gf}=(\mathbb {H} ,\Phi )$ as defined above. In the case of a commutative group, its dual commutative Hopf algebra is obtained by Fourier transformation of its dual finite Abelian quantum group elements.

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References

↑ ABE, E., Hopf Algebras , Cambridge University Press, 1977.
↑ SWEEDLER, M.E., Hopf Algebras , W.A. Benjamin, inc., New York, 1969.
↑ KUSTERMANS, J., VAN DAELE, A., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Int. J. of Math. 8 (1997), 1067-1139.
↑ LANCE, E.C., An explicit description of the fundamental unitary for $SU(2)_{q}$ , Commun. Math. Phys. 164 (1994), 1-15.

[Abe1977-1] ABE, E., Hopf Algebras , Cambridge University Press, 1977.

[SME1969-2] SWEEDLER, M.E., Hopf Algebras , W.A. Benjamin, inc., New York, 1969.

[KJVD1997-3] KUSTERMANS, J., VAN DAELE, A., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Int. J. of Math. 8 (1997), 1067-1139.

[LEC1994-4] LANCE, E.C., An explicit description of the fundamental unitary for $SU(2)_{q}$ , Commun. Math. Phys. 164 (1994), 1-15.

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