Ungar, Abraham A. Extension of the unit disk gyrogroup into the unit ball of any real inner product space. (English) Zbl 0865.20055 J. Math. Anal. Appl. 202, No. 3, 1040-1057 (1996). The author generalizes his example of a gyrogroup (= \(k\)-loop = Bruck loop), which he has defined previously [Aequationes Math. 47, No. 2-3, 240-254 (1994; Zbl 0799.20032)] in the open disc \(D_c:=\{x\in\mathbb{C}\mid|x|<c\}\), \(c>0\) by \(x\oplus y:=(x+y)c^2\cdot(c^2+\overline xy)^{-1}\), for the case that \(D_c\) is replaced by an open ball \(V_c:=\{{\mathfrak x}\in V\mid|{\mathfrak x}|<c\}\) of a real inner product space \((V,\mathbb{R},\cdot)\). He shows that the Möbius transformations \(a^\oplus:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\}\); \(x\to(a+x)c^2\cdot(c^2+\overline ax)^{-1}\), \(a\in D_c\) which serve for the definition of the loop operation “\(\oplus\)”, can also be defined for \(V\cup\{\infty\}\), and so \(V_c\) turned in a loop. The author studies the group generated by these “generalized Möbius transformations” which can be considered as the motion group of the hyperbolic space defined in \(V_c\) by introducing the “generalized Poincaré metric” \(d({\mathfrak x},{\mathfrak y}):=|{\mathfrak x}\ominus{\mathfrak y}|\). Reviewer: H.Karzel (München) Cited in 15 Documents MSC: 20N05 Loops, quasigroups 46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) 20E22 Extensions, wreath products, and other compositions of groups 22E43 Structure and representation of the Lorentz group 30A05 Monogenic and polygenic functions of one complex variable 83A05 Special relativity Keywords:generalized Poincaré metric; gyrogroups; \(k\)-loops; open balls; real inner product spaces; Möbius transformations; motion groups; hyperbolic spaces; Bruck loops Citations:Zbl 0799.20032 PDFBibTeX XMLCite \textit{A. A. Ungar}, J. Math. Anal. Appl. 202, No. 3, 1040--1057 (1996; Zbl 0865.20055) Full Text: DOI