Reductive group

The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.

For example, the simply connected split simple groups over a field k corresponding to the "classical" Dynkin diagrams are as follows:

As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the classical groups are: