affine algebraic group

An affine algebraic group over a field $k$ is quasi-affine variety $G$ (a locally closed subset of affine space) over $k$, which is a equipped with a group such that the multiplication map $m:G\times G\to G$ and inverse map $i:G\to G$ are algebraic.

For example, $k$ is an affine algebraic group over itself with the group law being addition, and as is $k^{*}=k-\{0\}$ with the group law multiplication. Other common examples of affine algebraic groups are ${\mathrm{GL}}_{n}k$, the general linear group over $k$ (identifying matrices with affine space) and any algebraic torus over $k$.