Compact Lie Groups
As we have seen in Chapter 5, Levi’s Theorem 5.6.6 is a central result in the structure theory of Lie algebras. It often allows splitting problems: one separately considers solvable and semisimple Lie algebras, and one puts together the results for both types. Naturally, this strategy also works to some extent for Lie groups. After dealing with nilpotent and solvable Lie groups in Chapter 11, we turn to the other side of the spectrum, to groups with semisimple or reductive Lie algebras. Here an important subclass is the class of compact Lie groups and the slightly larger class of groups with compact Lie algebra. Many problems can be reduced to compact Lie groups, and they are much easier to deal with than noncompact ones. The prime reason for that is the existence of a finite Haar measure whose existence was shown in Section 10.4.
KeywordsCompact Group Weyl Group Maximal Torus Splitting Theorem Cartan Subalgebras
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