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Journal of Lie Theory 25 (2015), No. 3, 787--805
Copyright Heldermann Verlag 2015

Three-Dimensional Topological Loops with Nilpotent Multiplication Groups

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

Margherita Lattuca
Dip. di Fisica e Chimica, UniversitÓ degli Studi di Palermo, 91023 Palermo, Via Archirafi 36, Italy

We describe the structure of indecomposable nilpotent Lie groups which are multiplication groups of three-dimensional simply connected topological loops. In contrast to the 2-dimensional loops there is no connected topological loop of dimension greater than 2 such that the Lie algebra of its multiplication group is an elementary filiform Lie algebra. We determine the indecomposable nilpotent Lie groups of dimension less or equal to 6 and their subgroups which are the multiplication groups and the inner mapping groups of the investigated loops. We prove that all multiplication groups have a 1-dimensional centre and the corresponding loops are centrally nilpotent of class 2.

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

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

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