What is an Example of a Co-commutative but not Commutative Quantum Group?

Question

I am looking for 'a' right candidate for an "abelian" quantum group.

In a comment to another question it was suggested that the correct candidate was co-commutative.

It is straightforward to show that if $G$ is an abelian group then $F(G)$ is co-commutative.

I had wanted to show that if $A$ was a co-commutative quantum group that could be realised as an algebra of functions on a group, that this group would be abelian and thus a conclusion could be made.

The first thing I did was find a nice structure theorem which I think (haven't looked through all the details) is that if $B$ is a commutative quantum group then $B\cong F(\Phi(B))$, where $\Phi$ are the characters on $B$.

I was worried for a moment that perhaps I was missing out on things that were co-commutative but not commutative... but of course these were the very things I was looking to make a definition for!

So now I am looking for an example of something that is non-commutative, so is not the algebra of functions on a group, but there is still a symmetry that behaves like "abelian-ness" up in the algebra morryah is co-commutative.

EDIT: I have a feeling that for my structure theorem to work the algebra must be finite dimensional... which rules out a few interesting options.

Answer 1

The standard example you will find in textbooks is the dual of a countable commutative group.

Answer 2

$\newcommand{\g}{\mathfrak{g}}$I don't work on quantum groups myself, but from what I understand, the two canonical examples in the quantum groups literature of noncommutative but cocommutative Hopf algebras are the group algebra $k[G]$ of a non-abelian group $G$ and the universal enveloping algebra $U(\g)$ of a Lie algebra $\g$. Indeed, it is a theorem of Kostant's that if $H$ is a cocommutative Hopf algebra over $\mathbb{C}$, then $$ H \cong k[G(H)] \# U(\operatorname{Prim}(H)), $$ where $$ G(H) = \{h \in H \mid h \neq 0, \;\Delta(h) = h \otimes h\} $$ is the group of group-like elements in $H$, and $$ \operatorname{Prim}(H) = \{h \in H \mid \Delta(h) = h \otimes 1 + 1 \otimes h\} $$ is the Lie algebra of primitive elements in $H$.

If you specifically want to work with quantum groups à la Woronowicz or Van Daele, perhaps the reduced group $C^\ast$-algebra of a non-abelian group should give you the example you want to keep in mind?