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Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normalintegral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, weclassify all finite dimensional representations $V$ of the fiber $G_K$ of $G$over $K:=\text{Frac}(R)$ with the property that the set of lattices of $V$ withrespect to $R$ which are $G$-modules is as well the set of lattices of $V$ withrespect to $R$ which are $\text{Lie}(G)$-modules. We apply this classificationto get a general criterion of extensions of homomorphisms between reductivegroup schemes over $\text{Spec} K$ to homomorphisms between reductive groupschemes over $\text{Spec} R$. We also show that for a simply connectedsemisimple group scheme over a reduced $\mathbb Q$--algebra, the category ofits representations is equivalent to the category of representations of its Liealgebra.

On Lie algebra modules which are modules over semisimple group schemes