# One question on map algebras

1- Suppose, $X$ be an scheme and $g$ an finite-dimensional Lie algebra over an algebraically closed field $k$. We denote by $M(X,g)$, the Lie algebra of regular functions on $X$ with values in $g$ (equivalently, the set of morphisms of schemes $X\to g‎$). If $A = ‎‎\mathcal{O}‎‎_X(X)$, then proof that there is an isomorphism $M(X,g)≅g⊗A$ of Lie algebras over $A$ and hence also over $k$.
I would define the Lie algebra of regular functions on $X$ with values in $\mathfrak{g}$ to be $\mathfrak{g} \otimes k[X]$ (assuming that the base field is $k$). What definitions are you using? –  Qiaochu Yuan Jan 4 '13 at 10:06
After your latest edit: the fact that $g$ is a Lie algebra is irrelevant now. View it just as a finite dimensional vector space. Suppose $X$ is an affine space: can you do this in that case? (now there are even less equivariant things in the question... you should fix the title!) –  Mariano Suárez-Alvarez Jan 28 '13 at 8:20