Exercise 9, § 4

Let $G$ be a compact Lie group over $K$, of dimension $n$. We suppose that the Lie algebra $L(G)$ is simple. For every $g \in G$, every integer $m \geq 1$, and every neighbourhood $V$ of $e$ in $G$, we denote by $M(g, m, V)$ the set of elements of $G$ of the form $\prod_{i=1}^{m} x_i(y_i, g)x_i^{-1}$, for $x_1, \ldots, x_m, y_1, \ldots, y_m$ in $V$.

a) Prove that if $g$ is an element of $G$ whose centralizer is not open, if $m$ is an integer $\geq n$, and if $V$ is a neighbourhood of $e$, then $M(g, m, V)$ is a neighbourhood of $e$. (There exists $a \in L(G)$ such that (Ad g)(a) ≠ a. Let φ be an exponential mapping of G. Let y be the image of 1 by the tangent mapping at 0 to $\lambda \mapsto (\varphi(\lambda a), g)$ (where $\lambda \in K$). We have $b = (\mathrm{Ad}\ g - 1)(a) \neq 0$. There exist $x_1, \ldots, x_m$ in V such that { (Ad $x_1$)(b), ..., (Ad $x_m$)(b) } contains a basis of L(G).

We consider the mapping $(x'_1, \ldots, x'_m, y_1, \ldots, y_m) \mapsto \prod_{i=1}^{m} x'_i(y_i, g){x'_i}^{-1}$ from $G^{2m}$ into G. Prove that the tangent mapping at $(x_1, \ldots, x_m, e, \ldots, e)$ is surjective.)

b) Let U be a neighbourhood of e in G. Let m be an integer $\geq 1$. There exists an element g of G whose centralizer is not open, such that M(g, m, G) ⊂ U. (Reason by contradiction using the compactness of G.)

c) Let G’ be a compact Lie group over K whose Lie algebra is simple. Let $\varphi : G \to G'$ be a bijective homomorphism of abstract groups. Then $\varphi$ is continuous. (Apply b) to G’ and a) to G.)