On Compact Abelian Lie Groups of Homeomorphisms of R^m

Khadija Ben Rejeb¹

¹Higher Institute of Computer Science and Communication Technologies, University of Sousse, Hammam Sousse, Tunisia

Journal of Lie Theory, Tome 31 (2021) no. 1, pp. 233-236

Résumé

Let $G$ be a compact Lie group of homeomorphisms of $\mathbb R^m$. The Naive conjecture saying that $G$ is conjugate to a subgroup of the orthogonal group $O(m)$ is known to be false for higher dimension. In this paper we give a partial answer by considering the action of the group $S = S(K_1) \times ... \times S(K_q)$ on $\mathbb R^m = K_1 \oplus ... \oplus K_q$, where $K_i = \mathbb R$ or $\mathbb C$ and $S(K_i) = \{x \!\in\! K_i : |x| = 1\}$ for $1\! \leq\! i \!\leq\! q$, and we show that $G$ is contained in $S$ if and only if every element of $G$ centralizes~$S$.

Détail

Classification : 37B05, 57S05, 57S10, 54H20, 37B20
Mots-clés : Compact Lie group, homeomorphism of the Euclidean space R^m, conjugate, orthogonal group