TITLE: $GL_n(k)$ (General linear group over a algebraically closed field) as a affine variety?
QUESTION [5 upvotes]: In the context of linar algebraic groups, I read in my notes from the lecture that's already some while ago that $GL_n(k)$ is an algebraic variety because $GL_n=D(\det)$, $ \det \in k [ (X_{ij})_{i,j} ]$.
Now, $k$ is an algebraically closed field, $\det$ is the determinant and $k [ (X_{ij})_{i,j} ]$ are the polynomials in unkwnowns $X_{ij}$. But I cannot find what $D$ meant, maybe it's also a typo or uncommon notation -.-
How can I interpret $GL_n (k)$ as a variety?
(Or what does this $D$ stand for?)

REPLY [9 votes]: I think $D$ becomes a more common notation when working with schemes, but here $D(f)$ for $f \in k[X_1, \ldots, X_n]$ should mean $\{a \in k^n : f(a) \neq 0\}$. So it's the open set where $f$ does not vanish. The important thing is that $D(f)$ is isomorphic to the algebraic set defined by $fX_{n + 1} - 1$ in $k^{n + 1}$ via the map $(a_1, \ldots, a_n) \mapsto (a_1, \ldots, a_n, 1/f(a_1, \ldots, a_n))$.