Linearity of Lie Groups

Abstract

In this chapter, we will make a connection to the topic of the first chapters by characterizing the connected Lie groups which admit faithful finite-dimensional representations. Eventually, it turns out that these are precisely the semidirect products of normal simply connected solvable Lie groups with linearly real reductive Lie groups, where the latter ones are, by definition, groups with reductive Lie algebra, compact center and a faithful finite-dimensional representation. We complement this result by several other characterizations, e.g., in terms of linearizers or properties of a Levi decomposition. Moreover, we characterize the complex Lie groups which admit finite-dimensional holomorphic linear representations, thus completing the discussion from Chapter 15.