Journal of Mathematical Sciences, Vol. 137, No. 5, 2006 COVERINGS OF TOPOLOGICAL LOOPS P. T. Nagy and K. Strambach UDC 515.14 Abstract. In this paper, we show that in some cases, no proper covering of a locally compact group topologically generated by left translations of a topological loop can occur as the group topologically generated by left translations of a topological loop. CONTENTS 1. Introduction .............................................. 5098 2. Some Basic Notions ......................................... 5099 3. Covering of Loops Realized in Solvable Lie Groups ........................ 5101 4. Topological Loops Realized as Sections in Three-Dimensional Quasi-Simple Lie Groups . . . 5105 5. Decomposition of Covering Group Manifolds of PSL 2 (R) .................... 5110 6. Sections in the Covering Groups of PSL 2 (R) ........................... 5114 References ............................................... 5116 1. Introduction The theory of coverings of topological groups allows one to see immediately for which groups G a given group  G is a covering. For a given connected Lie group  G having a continuous, sharply transitive section  σ :  G/  H →  G with respect to a connected subgroup  H , it is, however, not clear in which groups G having  G as a covering group, there exists a sharply transitive section σ : G/H → G such that the loop  L corresponding to the section  σ is a covering of the loop L corresponding to σ. One of the purposes of this paper is to show the complexity with respect to this situation. There are groups  G such that for any section  σ :  G/  H →  G and any group G covered by  G, there exists a section σ : G/H → G in G such that the corresponding loop L is covered by the loop  L. We illustrate this for topological loops having a three-dimensional, non-Abelian, nilpotent Lie group as the group topologically generated by their left translations. But we also show that already for higher-dimensional nilpotent groups, the situation substantially changes. There are nilpotent Lie groups G of nilpotency class 2 whose universal coverings contain sharply transitive sections corresponding to loops  L such that G cannot be the group topologically generated by left translations of topological loops L having  L as the universal covering. Moreover, the other extreme situation can also occur. There are groups  G having differentiable sections  σ :  G/  H →  G such that the exponential image exp(T 1  σ(  G/  H )) of the tangent space T 1  σ(  G/  H ) at 1 ∈  G is contained in the image  σ(  G/  H ) of the section  σ for which there exists no loop having a group G covered by  G as the group topologically generated by left translations. Three-dimensional topological loops having the direct product of a one-parameter group and a two-dimensional affine group as the group topologically generated by their left translations serve as examples in these cases. Another purpose of this paper is to show that in some cases, no proper covering of a locally compact group topologically generated by left translations of a topological loop can occur as the group topologically generated by left translations of a topological loop. We illustrate this for the group G = PSL 2 (R) of Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004. 5098 1072–3374/06/1375–5098 c  2006 Springer Science+Business Media, Inc.