Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O\_G$ the structural sheaf of $G$.


Let $D\_e : O\_{G,e} \to k$ a derivation.


I would like to get directly (ie, without any consideration about the cotangent bundle, or some canonical isomorphisms...) a derivation $D : O\_G\to O\_G$ that extends $D\_e$, and which is compatible with the action of $G$. That is, I would like to get this derivation by the mean of the multiplication map : $m : G \times G \to G$, etc., etc.


I have guessed this question would not be difficult, and would only be a matter of technics, but I can't manage to do it.



A technical question about derivations of sheaves on group schemes
Your derivation at the origin is a map $\mathrm{Spec} k[\epsilon] / \epsilon^2 \rightarrow G$ whose restriction to $\epsilon = 0$ is the inclusion of the origin. This induces a map


$$G \times\_{\operatorname{Spec} k} \operatorname{Spec} k[\epsilon]/\epsilon^2 \rightarrow G \times G \rightarrow G$$


where the first map is the product with the map described above and the second is the multiplication map. Dually, the composed map gives a map of sheaves of algebras


$$\mathcal{O}\_G \rightarrow \mathcal{O}\_G[\epsilon] / \epsilon^2$$


which is the same thing as a derivation from $\mathcal{O}\_G$ to itself.


