On the Multiplication Groups of Three-DimensionalTopological Loops

摘要

We clarify the structure of nilpotent Lie groups which are multipli-cation groups of 3-dimensional 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 nilpo-tent Lie groups for all filiform groups F_(n+2) and F_(m+2) with n, m > 1, the direct　product F_(n+2) × R and the direct product F_(n+2)× FnZ 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 gen-erated by the left translations.