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Journal of Lie Theory 21 (2011), No. 2, 385--415
Copyright Heldermann Verlag 2011

On the Multiplication Groups of Three-Dimensional Topological Loops

Ágota Figula
Institute of Mathematics, University of Debrecen, P.O.B. 12, 4010 Debrecen, Hungary


\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.

Keywords: Multiplication group of loops, topological transformation group, filiform Lie group.

MSC: 57S20, 57M60, 20N05, 22F30, 22E25

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