Fully Reducible Subgroups of Algebraic Groups

In the theory of Lie groups one often encounters statements which can be asserted for connected groups on the strength of the appropriate analogue for Lie algebras, but which may be invalid for non-connected groups. A good example of this phenomenon is the theorem of Sophus Lie that a solvable linear Lie algebra (over a field of characteristic 0) and hence a connected solvable linear Lie group can be simultaneously triangularized (upon extending the ground of its algebraic closure). This powerful principle is not valid for general solvable linear groups. In fact, an examination of proofs of results that are valid for only connected Lie groups often reveals that Lie's theorem has been used in an essential way. On the other hand, there are some results which are deduced for coinected groups from their Lie algebras which continue to ring true when the hypothesis of connectedness is dropped. In this paper we obtain several results of such a type. The results are related for most part to properties of fully reducible groups of linear transformations. Our central result concerns a decomposition of algebraic groups which is closely related to the Wedderburn decomposition of an associative algebra into the semi-direct sum of a semi-simple subalgebra and the radical. Our decomposition for algebraic groups can be viewed as a result on group extensions. Any algebraic group is a finite extension of the connected component of its identity G,. To what extent is the extension splittable? Put in another way, how large a normal connected subgroup N can we find so that N admits a complementary subgroup? The answer in any particular case depends on the arithmetic properties of the ground field. Nevertheless for a general ground field of characteristic zero, something can still be asserted.