# Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper subgroup and $G$ is connected then necessarily $\dim H < \dim G$, which is neat enough. With that, the argument follows by observing that if there were no maximal tori, then for any torus $\Bbb{T} \subset G$ there would be another torus $\Bbb{T}^\prime \subset G$ containing $\Bbb{T}$ such that $\dim \Bbb{T} < \dim \Bbb{T}^\prime$, as tori are always connected subgroups. Thus, there exists a strictly increasing sequence $\left\{\Bbb{T}_n \right\}$ of tori such that $$\dim \Bbb{T_n} < \dim \Bbb{T_{n+1}}, \forall\ n \in \Bbb{N}.$$ Then the argument ends saying that this cannot be the case because this sequence will eventually hit $\dim G$. Now I wonder: this is for sure if $G$ is finite-dimensional, but it seems to me that the argument will fail if $\dim G = \infty$, so my questions are if I am correct with my thought and if it is also true for infinite-dimensional Lie groups that maximal tori do always exist.

## 1 Answer

Maybe you could use Zorn's lemma.

Let $G$ be a (possibly infinite dimensional) compact Lie group. Give a partial order all the tori in $G$ by inclusion relation.

If $\{\mathbb{T}_\alpha\}$ is any linearly ordered set of tori, then their union $\cup\mathbb{T}_\alpha$ is also a (possibly infinite dimensional) torus because it is an abelian subgroup of $G$. Therefore it is an upper bound for $\{\mathbb{T}_\alpha\}$. Thus, by Zorn's lemma, there is a maximal torus.