Algebraic Groups and Lie Groups with Few Factors

In the theory of locally compact topological groups, the aspects and notionsfrom abstract group theory have conquered a meaningful place from the beginning(see New Bibliography in [44] and, e.g. [41–43]). Imposing grouptheoreticalconditions on the closed connected subgroups of a topologicalgroup has always been the way to develop the theory of locally compactgroups along the lines of the theory of abstract groups.Despite the fact that the class of algebraic groups has become a classicalobject in the mathematics of the last decades, most of the attention was concentratedon reductive algebraic groups. For an affine connected solvable algebraicgroup G, the theorem of Lie–Kolchin has been considered as definitivefor the structure of G, whereas for connected non-affine groups, the attentionturns to the analytic and homological aspects of these groups, which arequasi-projective varieties (cf. [79, 80, 89]). Complex Lie groups and algebraicgroups as linear groups are an old theme of group theory, but connectedness ofsubgroups 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 complexanalysis (cf. [1, 7]).In these notes we want to include systematically algebraic groups, as wellas real and complex Lie groups, in the frame of our investigation. Althoughaffine algebraic groups over fields of characteristic zero are related to linearLie 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 differencesbetween algebraic groups over a field of characteristic p > 0 and over fields ofcharacteristic zero. (...)