Cartan Decomposition On The Lie Group Level
Let be a semisimple Lie group and its Lie algebra. Let be a Cartan involution on and let be the resulting Cartan pair. Let be the analytic subgroup of with Lie algebra . Then
- There is a Lie group automorphism with differential that satisfies .
- The subgroup of elements fixed by is ; in particular, is a closed subgroup.
- The mapping given by is a diffeomorphism.
- The subgroup contains the center of, and is compact modulo center, that is, is compact.
- The subgroup is the maximal subgroup of that contains the center and is compact modulo center.
The automorphism is also called global Cartan involution, and the diffeomorphism is called global Cartan decomposition.
For the general linear group, we get as the Cartan involution.
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