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.