Journal of Lie Theory

no. 2
On the Multiplication Groups of Three-Dimensional Topological Loops
Journal of Lie Theory, Volume 21 (2011) no. 2, pp. 385-415
\def\R{\mathbb{R}} We clarify the structure of nilpotent Lie groups which are multiplication groups of $3$-dimension\-al simply connected topological loops and prove that non-solvable Lie groups acting minimally on $3$-dimensional manifolds cannot be the multiplication group of $3$-dimensional topological loops. Among the nilpotent Lie groups for all filiform groups ${\cal F}_{n+2}$ and ${\cal F}_{m+2}$ with $n, m > 1$, the direct product ${\cal F}_{n+2} \times \R$ and the direct product ${\cal F}_{n+2} \times_Z {\cal F}_{m+2}$ with amalgamated center $Z$ occur as the multiplication group of $3$-dimensional topological loops. To obtain this result we classify all $3$-dimensional simply connected topological loops having a $4$-dimensional nilpotent Lie group as the group topologically generated by the left translations.
DOI: 10.5802/jolt.636
Classification: 57S20, 57M60, 20N05, 22F30, 22E25
Keywords: Multiplication group of loops, topological transformation group, filiform Lie group 
@article{JOLT_2011_21_2_a5,
     author = {A. Figula},
     title = {On the {Multiplication} {Groups} of {Three-Dimensional} {Topological} {Loops}},
     journal = {Journal of Lie Theory},
     pages = {385--415},
     year = {2011},
     volume = {21},
     number = {2},
     doi = {10.5802/jolt.636},
     zbl = {1222.22020},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.636/}
}
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A. Figula. On the Multiplication Groups of Three-Dimensional Topological Loops. Journal of Lie Theory, Volume 21 (2011) no. 2, pp. 385-415. doi: 10.5802/jolt.636

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