Maximal Subgroups of Compact Lie Groups

Fernando Antoneli¹ ; Michael Forger² ; Paola Gaviria²

¹Escola Paulista de Medicina, Universidade Federal de São Paulo, 04039-062 São Paulo - SP, Brazil
²Inst. de Matemática e Estatística, Universidade de São Paulo, PO Box 66281, 05315-970 São Paulo - SP, Brazil

Journal of Lie Theory, Tome 22 (2012) no. 4, pp. 949-1024

Résumé

This report 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 Σ-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Σ-primitive subalgebras of compact simple Lie algebras, where Σ is a subgroup of the corresponding outer automorphism group.
In the second part, we explicitly compute the normalizers of the primitive subalgebras 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.

Détail

Classification : 22E15
Mots-clés : Lie groups, Lie algebras, Compact groups, Maximal subgroups