Suppose $\mathfrak{g}$ is a Lie algebra, let $\mathfrak{g}^* $ be its dual. I need $\mathfrak{g}^* $ to be an affine algebraic variety. How is it one?
I suppose the only important thing is that $\mathfrak{g}^* $ is a vector space, and the additional structure doesn't matter. But even for vector spaces how I don't know how that works: I have read here that given a vector space $V$, one can construct an affine scheme by taking the spectrum of the symmetric algebra of its dual $V^* $. But in my case I have that $\mathfrak{g}^* $ itself is a scheme. Do I just fix a basis of $\mathfrak{g}^* $ so that $g^* $ is isomorphic to some $\mathbb{A}^n$ and then go from there?