%0 Generic %1 Gaviria2006 %A Gaviria, Paola %A Forger, Michael %A Antoneli, Fernando %D 2006 %K GroupTheory %T Maximal Subgroups of Compact Lie Groups %U http://arxiv.org/abs/math/0605784 %X In this article, we investigate the problem of classifying the maximal subgroups of a general compact Lie group $G$. First, when $G$ is not connected, it is shown how to reduce this problem to that of finding the maximal subgroups of the connected one-component $G_0$ of $G$ and the maximal subgroups of the finite group $G/G_0$. Then it is shown that the classification of the maximal subgroups of connected compact Lie groups may be reduced to the classification of their maximal finite subgroups together with the classification of the maximally invariant subalgebras of compact Lie algebras, whose normalizers define maximal subgroups. It is also shown how this second task may be further reduced to the special case of compact simple Lie algebras, whose maximally invariant subalgebras are identical with their primitive subalgebras. Finally, 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.

@misc{Gaviria2006, abstract = { In this article, we investigate the problem of classifying the maximal subgroups of a general compact Lie group $G$. First, when $G$ is not connected, it is shown how to reduce this problem to that of finding the maximal subgroups of the connected one-component $G_0$ of $G$ and the maximal subgroups of the finite group $G/G_0$. Then it is shown that the classification of the maximal subgroups of connected compact Lie groups may be reduced to the classification of their maximal finite subgroups together with the classification of the maximally invariant subalgebras of compact Lie algebras, whose normalizers define maximal subgroups. It is also shown how this second task may be further reduced to the special case of compact simple Lie algebras, whose maximally invariant subalgebras are identical with their primitive subalgebras. Finally, 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. }, added-at = {2009-03-15T14:55:09.000+0100}, author = {Gaviria, Paola and Forger, Michael and Antoneli, Fernando}, biburl = {https://www.bibsonomy.org/bibtex/29ca78693d62a0634556682c172acf39f/gber}, description = {Maximal Subgroups of Compact Lie Groups}, interhash = {9634baa0afd7353db8dc15adf9fef73f}, intrahash = {9ca78693d62a0634556682c172acf39f}, keywords = {GroupTheory}, note = {cite arxiv:math/0605784 Comment: 44 pages}, timestamp = {2009-03-15T14:55:09.000+0100}, title = {Maximal Subgroups of Compact Lie Groups}, url = {http://arxiv.org/abs/math/0605784}, year = 2006 }