I am using this definition:



> 
> An *algebra of functions on a finite quantum group* $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
> 
> 
> 


I have the following (very well known --- folklore --- result)



> 
> *(Classification Theorem)*
> 
> 
> Let $A$ be the algebra of functions on a finite quantum group
>  $\mathbb{G}$:
> 
> 
> 1. if $A$ is commutative then $\mathbb{G}\cong \Phi(A)$.
> 2. if $A$ is cocommutative then $A=F(\mathbb{G})\cong \mathbb{C} \Phi(A)=:F(\widehat{\Phi(A)})$.
> 
> 
> 


Here $\Phi(A)$ is the set of characters on $A$.


I want to reference these results but am struggling somewhat to find good, old, authoritative references. Any help would be appreciated.



Commutative and Cocommutative Quantum Groups
I took a quick look into Timmermann's book "An invitation to Quantum Groups".


It refers to:


Saad Baaj; Georges Skandalis
*Unitaires multiplicatifs et dualité pour les produits croisés de C\*-algèbres*
Annales scientifiques de l'École Normale Supérieure (1993)
Volume: 26, Issue: 4, page 425-488
ISSN: 0012-9593


They describe finite dimensional C${}^\*$-Hopf algebras in terms of multiplicative unitaries.
Theorem 2.2 shows that if you have commutative multiplicative unitaries you get a locally compact group and therefore in the finite case, a finite group.


This gives the first statement. The second statement is equivalent to the first by duality.


\*edit\*


Such a result is already stated as Theorem 3.3 in


L. I. Vaĭnerman and G. I. Kac, *Nonunimodular Ring Groups and Hopf-Von Neumann algebras*,
Mathematics of the USSR-Sbornik, Volume 23, Number 2


