Finite Quantum (Hopf) Algebra

Recall that:

Definition 1.1 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.

Definition 1.2 A finite quantum algebra $A_{Gf}$ is the dual of a finite quantum group

$\displaystyle Q_{Gf}=(\mathbb{H},\Phi)$

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.

Bibliography

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 , Commun. Math. Phys. 164 (1994), 1-15.

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