Topological Loops with Decomposable Solvable Multiplication Groups

Abstract

In this paper we deal with the class mathcal {C} of decomposable solvable Lie groups having dimension six. We determine those Lie groups in mathcal {C} and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.