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.
Salvatore Siciliano
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