Journal of Lie Theory

no. 1
On Compact Abelian Lie Groups of Homeomorphisms of Rm
Journal of Lie Theory, Volume 31 (2021) no. 1, pp. 233-236
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~$S$.
DOI: 10.5802/jolt.1167
Classification: 37B05, 57S05, 57S10, 54H20, 37B20
Keywords: Compact Lie group, homeomorphism of the Euclidean space Rm, conjugate, orthogonal group 
@article{JOLT_2021_31_1_a11,
     author = {K. Ben Rejeb},
     title = {On {Compact} {Abelian} {Lie} {Groups} of {Homeomorphisms} of {R\protect\textsuperscript{m}}},
     journal = {Journal of Lie Theory},
     pages = {233--236},
     year = {2021},
     volume = {31},
     number = {1},
     doi = {10.5802/jolt.1167},
     zbl = {1472.57034},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1167/}
}
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K. Ben Rejeb. On Compact Abelian Lie Groups of Homeomorphisms of Rm. Journal of Lie Theory, Volume 31 (2021) no. 1, pp. 233-236. doi: 10.5802/jolt.1167

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