1
$\begingroup$

Let $G$ be a non-discrete Lie group. Can the connected component of its identity ($G_e$) be presented as the union of its Lie connected subgroups of dimension 1: $$G_e = \bigcup\{ \text{Lie connected subgroups of } G \text{ of dimension } 1 \}?$$

$\endgroup$
4
  • 4
    $\begingroup$ No. 1 dimensional connected subgroups can be shown to be 1-parameter subgroups. So they must be in the image of $\exp$ and that is not surjective in general. $\endgroup$
    – Callum
    Commented Nov 7, 2023 at 13:40
  • $\begingroup$ @Callum What about those isomorphic to $\mathbb{S}^1$? $\endgroup$ Commented Nov 7, 2023 at 22:42
  • $\begingroup$ What about them? They are still 1parameter subgroups. $\endgroup$
    – Callum
    Commented Nov 8, 2023 at 7:17
  • $\begingroup$ @Callum Oh, I didn't know that they are also 1-parameter groups. $\endgroup$ Commented Nov 8, 2023 at 21:41

0

Browse other questions tagged .