Algebraic Groups and Lie Groups with Few Factors

Abstract

In the theory of locally compact topological groups, the aspects and notions from abstract group theory have conquered a meaningful place from the beginning (see New Bibliography in [44] and, e.g. [41–43]). Imposing grouptheoretical conditions on the closed connected subgroups of a topological group has always been the way to develop the theory of locally compact groups along the lines of the theory of abstract groups. Despite the fact that the class of algebraic groups has become a classical object in the mathematics of the last decades, most of the attention was concentrated on reductive algebraic groups. For an affine connected solvable algebraic group G, the theorem of Lie–Kolchin has been considered as definitive for the structure of G, whereas for connected non-affine groups, the attention turns to the analytic and homological aspects of these groups, which are quasi-projective varieties (cf. [79, 80, 89]). Complex Lie groups and algebraic groups as linear groups are an old theme of group theory, but connectedness of subgroups does not play a crucial rˆole in this approach, as can be seen in [97]. Non-linear complex commutative Lie groups are a main subject of complex analysis (cf. [1, 7]). In these notes we want to include systematically algebraic groups, as well as real and complex Lie groups, in the frame of our investigation. Although affine algebraic groups over fields of characteristic zero are related to linear Lie groups (cf. [11–13]), the theorems depending on the group topology differ (cf. e.g. Remark 5.3.6). For algebraic groups we want to stress the differences between algebraic groups over a field of characteristic p > 0 and over fields of characteristic zero. (...)