[math] \newcommand{\mathds}{\mathbb}[/math]

Let us call finite Hopf algebra a finite dimensional [math]C^*[/math]-algebra, with maps as follows, called comultiplication, counit and antipode,

[[math]] \Delta:A\to A\otimes A\quad,\quad \varepsilon:A\to\mathbb C\quad,\quad S:A\to A^{opp} [[/math]]

satisfying the following conditions, which are those found above,

[[math]] (\varepsilon\otimes id)\Delta=(id\otimes\varepsilon)\Delta=id [[/math]]

[[math]] m(S\otimes id)\Delta=m(id\otimes S)\Delta=\varepsilon(.)1 [[/math]]

along with [math]S^2=id[/math]. Prove that if [math]G,H[/math] are finite abelian groups, dual to each other via Pontrjagin duality, then we have an identification of Hopf algebras as follows,

[[math]] C(G)=C^*(H) [[/math]]

and based on this, go ahead and formally write any finite Hopf algebra as

[[math]] A=C(G)=C^*(H) [[/math]]

and call [math]G,H[/math] finite quantum groups, dual to each other.