Algebraic Groups

The definition of an algebraic group is similar to that of a Lie group, except that differentiable manifolds are replaced by algebraic varieties and differentiable maps by morphisms of algebraic varieties. In this book we will only consider the algebraic groups whose underlying varieties are affine ones. They are called “affine” or “linear” algebraic groups. The difference between arbitrary groups and affine ones is quite essential from the point of view of algebraic geometry and almost indiscernible from the group-theoretical points of view, since the commutator group of any irreducible algebraic group is an affine algebraic group. Besides, the general linear groups and any of their algebraic subgroups are affine algebraic groups. Therefore the affine algebraic groups are the most interesting ones for the Lie group theory. We will simply call them algebraic groups.