affine algebraic group An affine algebraic group is an affine algebraic variety together with a specified group structure (on the set of elements of ) such that group multiplication and inversion are morphisms of affine algebraic varieties. Morphisms For and affine algebraic groups, a map is a called a morphism of affine algebraic groups (or just a morphism) if is a morphism of affine algebraic varieties. is a group homomorphism.