Extension of the unit disk gyrogroup into the unit ball of any real inner product space. (English) Zbl 0865.20055

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” \(d({\mathfrak x},{\mathfrak y}):=|{\mathfrak x}\ominus{\mathfrak y}|\).


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


Zbl 0799.20032
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