Finite Quantum (Hopf) Algebra

Recall that:

Definition 1.1   A finite quantum group is a pair of a finite-dimensional -algebra with a comultiplication such that is a Hopf -algebra.

Definition 1.2   A finite quantum algebra is the dual of a finite quantum group

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.