Extension of the unit disk gyrogroup into the unit ball of any real inner product space
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'' \(d({\mathfrak x},{\mathfrak y}):=|{\mathfrak x}\ominus{\mathfrak y}|\).
scientific article; zbMATH DE number 934222