# Map algebras between scheme and Lie algebra

Suppose, $X$ be an arbitrary scheme over an algebraically closed field $k$.
1- In general, what is the structure of $A= \mathcal{O}_X(X)$?
2- If $g$ be an finite-dimensional Lie algebra over k. Then $g$ is naturally equipped with the structure of an affine algebraic scheme. Define $M(X,g)$ be the Lie algebra of regular functions on $X$ with values in $g$ (equivalently, the set of morphisms of schemes $X \rightarrow g$), then proof that there is an isomorphism $M(X,g) \cong g⊗A$ of Lie algebras over $A$ and hence also over $k$.
1) Look at the affine case, this already tells you that your question is way too broad. On the other hand, if $X/k$ is proper and connected, then $A=k$. –  Martin Brandenburg Jan 31 '13 at 17:07