Cartan's theorem (Lie groups)

In mathematics , Cartan's theorem , also known as the Closed Subgroup Theorem in English-language literature, states that closed subgroups of a Lie group are embedded submanifolds and, in particular, sub-Lie groups. It was proven in 1930 by Élie Cartan and, for matrix groups, in 1929 by John von Neumann . It is important for the classification of linear groups and for the construction of homogeneous spaces .

Explanations and examples

Torus

A subgroup of a Lie group does not necessarily have to be closed in the topology of the Lie group. For example the subgroups of the torus

${\displaystyle G=\mathbb {T} ^{2}=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right)\right|\theta ,\phi \in \mathbb {R} \right\}}$

from the shape

${\displaystyle H_{\alpha }=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\alpha \theta }\end{matrix}}\right)\right|\theta \in \mathbb {R} \right\}}$

for a . While one obtains for a closed subgroup , for irrational numbers the subgroups lie close in and in particular are not closed. ${\displaystyle \alpha \in \mathbb {R} }$ ${\displaystyle \alpha \in \mathbb {Q} }$ ${\displaystyle H_{\alpha }\subset G}$ ${\displaystyle \alpha \not \in \mathbb {Q} }$ ${\displaystyle H_{\alpha }}$ ${\displaystyle G}$

If the subgroup is not terminated in , then the subspace topology of generated by the topology of does not match the Lie group topology of . In the above example, the subgroups are abstractly (as groups) isomorphic to , but the topology induced by the torus does not match the topology of the Lie group . In contrast, for the closed subgroups, also as Lie groups, are isomorphic to the circle group . ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle \alpha \not \in \mathbb {Q} }$ ${\displaystyle H_{\alpha }}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \alpha \in \mathbb {Q} }$ ${\displaystyle H_{\alpha }}$ ${\displaystyle S^{1}}$

Cartan's theorem : A subgroup of a Lie group is an embedded sub-Lie group if and only if it is closed. ${\displaystyle H}$ ${\displaystyle G}$

literature

• John von Neumann: About the analytical properties of groups of linear transformations and their representations. Math. Z. 30 (1929), no. 1, 3-42.
• Elie Cartan: La théorie des groupes finis et continus et l'Analysis Situs. Mémorial Sc. Math. XLII, pp. 1-61 (1930).