It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is **not** a Hopf subalgebra?



Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra
The Hopf algebra $H = k[x^{\pm 1}]$ with $\Delta(x) = x \otimes x$ is the group algebra of $\mathbb{Z}$, the free group on one generator. Its subalgebra $k[x]$ is the "monoid algebra" for the submonoid $\mathbb{N} \subset \mathbb{Z}$, the free monoid on one generator. $k[x] \subset k[x^{\pm 1}]$ is not a Hopf subalgebra just because $\mathbb{N}$ is not a group.


(Note that $k[x]$ does have a natural Hopf structure with $\Delta(x) = x\otimes 1 + 1 \otimes x$, but it is not the one that extends along $k[x] \subset k[x^{\pm1}]$.)


