Commutative and Cocommutative Quantum Groups 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
  $\mathbb{G}$:
  
  
*
  
*if $A$ is commutative then $\mathbb{G}\cong \Phi(A)$.
  
*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.
 A: 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
A: I do not work in this area, but take a look at this survey article in case it gives some directive to your question.
A: If the definition of a finite quantum group, you use, is a pair $(A,\Phi)$ of a finite dimensional $C^*$-algebra $A$, with a comultiplication $\Phi$, such that $(A,\Phi)$ is a Hopf $*$-algebra, then this paper may be helpful. 
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