What is an example of a Hopf algebra $(H,\Delta,\epsilon)$ containing an invertible element $h$ which is not grouplike: An element $h \in H$ such that $$ \Delta(h) \neq h \otimes h\qquad\text{(not grouplike)} $$ and such that there exists a $h^{-1} \in H$ with $hh^{-1} = h^{-1}h = 1$ (invertible).

## 2 Answers

Let $L$ be a finite-dimensional $p$-nilpotent restricted Lie algebra over a field of characteristic $p>0$ and consider its restricted enveloping algebra $u(L)$. Then the only group-like element of the Hopf algebra $u(L)$ is 1. On the other hand, every element of $u(L)$ that is not in the kernel of the counit is invertible.

Let $G$ be any finite group. Then the group algebra $\mathbb{C}[G]$ is, as an algebra, isomorphic to $\bigoplus_V \mathrm{End}(V)$, where the direct sum is over irreducible representations of $V$ of $G$. So the group of units of $\mathbb{C}[G]$ is isomorphic to $\prod_V \mathrm{GL}(V)$. Only finitely many elements of this product come from the original group $G$. Any element of $\prod_V \mathrm{GL}(V)$ not coming from $G$ is invertible but not group like.

To be concrete, let $G$ be the cyclic group $C_2$, with nontrivial element $g$. Then $a+bg$ is invertible as long as $(a+b)(a-b) \neq 0$, but $a+bg$ is group like only for $(a,b) = (1,0)$ and $(a,b) = (0,1)$.

1more comment