Linear algebraic group
One of the first uses for the theory was to define the Chevalley groups.
Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,
One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group G written as an extension
that satisfies the axioms of a group action. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.
There are several reasons why a Lie group may not have the structure of a linear algebraic group over R.