affine algebraic group

An affine algebraic group is an affine algebraic variety $G$ together with a specified group structure (on the set of elements of $G$) such that group multiplication $m:G\times G⟶G$ and inversion $i:G⟶G$ are morphisms of affine algebraic varieties.

Morphisms

For $G$ and $H$ affine algebraic groups, a map $\phi :G\to H$ is a called a morphism of affine algebraic groups (or just a morphism) if

1. $\phi$ is a morphism of affine algebraic varieties.
2. $\phi$ is a group homomorphism.