Algebraic Subgroups in Lie Groups

In this section we shall examine the question of when a complex Lie group G may be given the structure of an affine algebraic group which is compatible with its analytic group structure in the sense that the rational representations of G are exactly the complex analytic representations of G. To answer this question in a more general context, we introduce and study the notion of algebraic subgroups in a complex Lie group ([12], [23], [24]).