Is a field a Hopf algebra? I'm not sure but I think that a field is a Hopf algebra.
2026-03-28 05:00:41.1774674041
Is a field a Hopf algebra?
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When you talk about Hopf algebras you have fixed a ground field $k$ and you are really talking about Hopf $k$-algebras.
Now a Hopf $k$-algebra $H$ comes with a counit $\epsilon:H\to k$ which is a map of $k$-algebras. If $H$ is a field, this counit must be injective and, since it is always surjective, it is an isomorphism.
This tells us that the unique Hopf $k$-algebra which is a field is $k$ itself.