On local parameters at the origin in an algebraic group 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 $k[G]$?
 A: If $A$ were always a sub-Hopf-algebra then its spectrum would be an algebraic group $H$ and there would be a non-constant map from $G$ to $H$. Furthermore, as an algebraic variety, $H$ would be isomorphic to affine $n$-space for some $n$ (equal to the dimension of $G$). Often this cannot happen. For example if $G$ is the multiplicative group, which would have been the first example I'd tried if I'd wanted to get my hands dirty and actually do the calculation, then the kernel of $G\to H$ would have to be finite, and the image would be an open dense subgroup and hence the whole thing. But the multiplicative group is not isomorphic to affine 1-space as an algebraic variety.
A: 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.
