finite quantum algebra

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 $ SU(2)_q$, Commun. Math. Phys. 164 (1994), 1-15.



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