Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ t\_1, \ldots, t\_n \} \subseteq I$ be a set of local parameters at $e$. Since $e$ is a smooth point, the $t\_i$ are algebraically independent and we naturally obtain a $k$-algebra embedding $k[t\_1, \ldots, t\_n] \hookrightarrow k[G]$. Let $A$ denote the subalgebra of $k[G]$ thus defined.


I do not expect that $A$ will be a sub-Hopf algebra of $k[G]$ (in particular, I see no reason why the comultiplication should preserve $A$), but I don't know any concrete examples. So I have two questions: (1) Is there a specific example of an algebraic group $G$ as above such that $A$ is not a sub-Hopf algebra of $k[G]$? (2) Are there nice conditions on $G$ under which $A$ will be a sub-Hopf algebra of $k[G]$?



On local parameters at the origin in an algebraic group
This *never* happens for a reductive group $G$ of dimension $n>0$. The conditions you give induce a surjective finite group homomorphism $G\to \mathbb{A}^n$, for some mysterious group structure on $\mathbb{A}^n$. In particular, $G$ would act transitively on $\mathbb{A}^n$, which a reductive group cannot do.


On the other hand, there are many unipotent groups whose underlying schemes are affine spaces, e.g. unipotent upper triangular matrices. Any extension of such a group by a finite group will satisfy the conditions of your question.


