Commutative Hopf algebra

A Hopf algebra is commutative when its multiplication is commutative. This is independent of whether its coproduct is cocommutative; function algebras of finite nonabelian groups give commutative but noncocommutative examples.

Functions on a finite group form a Hopf algebra under pointwise multiplication, with $(\Delta f)(g,h)=f(gh)$, $\epsilon (f)=f(1)$ and $(Sf)(g)=f(g^{-1})$. The coproduct is cocommutative exactly when the group is abelian, since delta functions distinguish distinct products $gh,hg$.