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Hugo Chapdelaine
Hugo Chapdelaine
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Iwasawa decomposition for homotopy type of connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:

(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compact subgroup of $G$. Moreover, all maximal compact subgroups of $G$ are conjugate to $K$.

So inOne nice corollary of this is that the group $K$ is a deformation retract of $G$ so these two groups have the same homotopy type. In the case where $G$ is semi-simple one may deduce the first part of the result above using the existence of the Iwasawa decomposition which is not so trivial to prove! But here since I'm only asking for a diffeomorphism between two manifolds, there might be a simpler argument!

Iwasawa decomposition for connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:

(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compact subgroup of $G$. Moreover, all maximal compact subgroups of $G$ are conjugate to $K$.

So in the case where $G$ is semi-simple one may deduce the first part of the result above using the existence of the Iwasawa decomposition which is not trivial to prove! But here since I'm only asking for a diffeomorphism between two manifolds, there might be a simpler argument!

homotopy type of connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:

(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compact subgroup of $G$. Moreover, all maximal compact subgroups of $G$ are conjugate to $K$.

One nice corollary of this is that the group $K$ is a deformation retract of $G$ so these two groups have the same homotopy type. In the case where $G$ is semi-simple one may deduce the first part of the result above using the existence of the Iwasawa decomposition which is not so trivial to prove! But here since I'm only asking for a diffeomorphism between two manifolds, there might be a simpler argument!

Source Link
Hugo Chapdelaine
Hugo Chapdelaine
  • 7.6k
  • 2
  • 28
  • 70

Iwasawa decomposition for connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:

(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compact subgroup of $G$. Moreover, all maximal compact subgroups of $G$ are conjugate to $K$.

So in the case where $G$ is semi-simple one may deduce the first part of the result above using the existence of the Iwasawa decomposition which is not trivial to prove! But here since I'm only asking for a diffeomorphism between two manifolds, there might be a simpler argument!