Projectivity of homogeneous left loops on Lie groups. I: Algebraic framework. (English) Zbl 0703.22003

Let (G,\(\mu\)) be a left loop with the unit e and the associated ternary operation \(\eta\) given by \(\eta (x,y,z)=L_ x\mu (L_ x^{-1}y,L_ x^{-1}z),\) where \(L_ x\) is the left translation by x in G. The loop (G,\(\mu\)) is homogeneous if it has the left inverse property: \(L_ x^{-1}=L_{x^{-1}}\) for \(x^{-1}=L_ x^{-1}e\) and any left mapping \(L_{x,y}=L_{\mu (x,y)}L_ xL_ y\) is an automorphism of (G,\(\mu\)). For two abstract homogeneous loops (G,\(\mu\)) and (G,\({\tilde \mu}\)) given on the same underlying set G the notion of projective relation is introduced. All homogeneous left loops which are in projective relation with an arbitrary given abstract group \((G,\mu^ 0)\) are investigated and a necessary and sufficient condition for the multiplication \({\tilde \mu}\) on G is found under which (G,\({\tilde \mu}\)) be in projective relation with the group \((G,\mu^ 0)\). This result generalizes a similar result which the author obtained in the case when the given group is the additive group on \({\mathbb{R}}^ n\) [see the author, Mem. Fac. Sci., Shimane Univ. 22, 33-41 (1988; Zbl 0698.22007)]. This algebraic result will be applied by the author to the study of homogeneous left loops on Lie groups.
Reviewer: V.V.Goldberg


22A30 Other topological algebraic systems and their representations
20N05 Loops, quasigroups
53A60 Differential geometry of webs
22E05 Local Lie groups
17B99 Lie algebras and Lie superalgebras


Zbl 0698.22007