Semisimple Group

Classification of split reductive groups

More generally, the classification of split reductive groups is the same over any field. A semisimple group G over a field k is called simply connected if every central isogeny from a semisimple group to G is an isomorphism. (For G semisimple over the complex numbers, being simply connected in this sense is equivalent to G(C) being simply connected in the classical topology.) Chevalley's classification gives that, over any field k, there is a unique simply connected split semisimple group G with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of adjoint type if its center is trivial. The split semisimple groups over k with given Dynkin diagram are exactly the groups G/A, where G is the simply connected group and A is a k-subgroup scheme of the center of G.