Equivalence of two gyrogroup structures on unit balls
The concepts gyrogroup and \(K\)-loop are the same and, as \textit{A. Kreuzer} [Beitr. Geom. Algebra 33, 15-22 (1995)]\ has proved recently, any \(K\)-loop \((F,+)\) is a Bruck-loop, i.e. \(\forall a,b,c\in F:a+(b+(a+c))=(a+(b+a))+c\) (Bol identity) and \(-(a+b)=-a+(-b)\). \textit{A. Ungar} presented in 1990 and 1991 [Result. Math. 17, No. 3/4, 342-354 (1990; Zbl 0706.53039) and Am. J. Phys. 59, 829-834 (1991)]\ two examples of gyrogroups. In this paper it is shown that these loops are isomorphic. For this purpose concepts of homomorphism and isomorphism for gyrogroups are defined in such a way that, if \((F,+)\) and \((G,+)\) are isomorphic, then also the groups \((F,+)\rtimes_Q\text{Aut}(F,+)\) and \((G,+)\rtimes_Q\text{Aut}(G,+)\) obtained by the quasidirect product are isomorphic in the ordinary sense. For the considered examples \(\text{Aut}(F,+)\) is the orthogonal group \(O(n)\) and \((F,+)\rtimes_Q\text{Aut}(F,+)\) the Lorentzian group of the \(n\)-dimensional Minkowski world.
scientific article; zbMATH DE number 828281