Our study of real semisimple Lie groups and algebras is based on the theory of complex semisimple Lie groups developed in Ch. 4. This is possible because the complexification of a real semisimple Lie algebra is also semisimple (see 1.4.7). However, the correspondence between real and complex semisimple Lie algebras established with the help of the complexification is not one-to-one; any complex semisimple Lie group has at least two non-isomorphic real forms. As it turns out, to describe the real forms of a given complex semisimple Lie algebra g is the same as to classify the involutive automorphisms of g up to conjugacy in Aut g. This classification is easily obtained from the results of 4.4. The global classification of real semisimple Lie groups makes use of the so-called Cartan decomposition of these groups which also plays an important role in various applications of the Lie group theory.
Keywords
- Real Form
- Maximal Compact Subgroup
- Cartan Decomposition
- Compact Real Form
- Real Algebraic Group
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