
Journal of Lie Theory 21 (2011), No. 2, 385415 Copyright Heldermann Verlag 2011 On the Multiplication Groups of ThreeDimensional Topological Loops Ágota Figula Institute of Mathematics, University of Debrecen, P.O.B. 12, 4010 Debrecen, Hungary figula@math.klte.hu [Abstractpdf] \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 nonsolvable 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 [ Fulltextpdf (388 KB)] for subscribers only. 