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On second thought, the theorem 1 which was linked is probably not useful when specifically considering subgroups endowed with the induced topology.
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Samuel Monnier
Samuel Monnier
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Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?

Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.

I am aware that in finite dimensions, Cartan's theorem ensures that any closed subgroup is a Lie group. See also Theorem 1 here for a refinement of this condition.

In Neeb's notes about infinite dimensional Lie groups, it is mentioned that already for Banach-Lie groups, Cartan's theorem is not true anymore. However locally compact subgroups are Lie subgroups, see p.59.

Is there anything known beyond this?

Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?

Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.

I am aware that in finite dimensions, Cartan's theorem ensures that any closed subgroup is a Lie group. See also Theorem 1 here for a refinement of this condition.

In Neeb's notes about infinite dimensional Lie groups, it is mentioned that already for Banach-Lie groups, Cartan's theorem is not true anymore. However locally compact subgroups are Lie subgroups, see p.59.

Is there anything known beyond this?

Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?

Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.

I am aware that in finite dimensions, Cartan's theorem ensures that any closed subgroup is a Lie group.

In Neeb's notes about infinite dimensional Lie groups, it is mentioned that already for Banach-Lie groups, Cartan's theorem is not true anymore. However locally compact subgroups are Lie subgroups, see p.59.

Is there anything known beyond this?

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Samuel Monnier
Samuel Monnier
  • 1.6k
  • 12
  • 18

When is a subgroup of a Lie group itself a Lie group?

Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?

Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.

I am aware that in finite dimensions, Cartan's theorem ensures that any closed subgroup is a Lie group. See also Theorem 1 here for a refinement of this condition.

In Neeb's notes about infinite dimensional Lie groups, it is mentioned that already for Banach-Lie groups, Cartan's theorem is not true anymore. However locally compact subgroups are Lie subgroups, see p.59.

Is there anything known beyond this?