Mathematics – Rings and Algebras
Scientific paper
2006-05-31
Mathematics
Rings and Algebras
83 pages. Substantial changes on sections 2-5 in order to work only with closed subgroups
Scientific paper
This survey aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to (2) the classification of the finite maximal $\Sigma$-invariant subgroups of center-free connected compact simple Lie groups and (3) the classification of the $\Sigma$-primitive sub-algebras of compact simple Lie algebras, where $\Sigma$ is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive sub-algebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups.
Antoneli Fernando
Forger Michael
Gaviria Paola
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