Group-like structure underlying the unit ball in real inner product spaces (Q2639179)

In 1965 the reviewer introduced the notion near-domain \((F,+,\cdot)\) [cf. Abh. Math. Semin. Univ. Hamb. 32, 191-206 (1968; Zbl 0162.241)] i.e. (F1) \((F,+)\) is a loop (with identity 0) with \(``a+b=0\Rightarrow b+a=0''\), (F2) \((F^*,\cdot)\) is a group where \(F^*=F\setminus \{0\}\), (F3) \(\forall a,b,c\in F:\) \(a(b+c)=a\cdot b+a\cdot c\) and (F4) \(\exists d_{a,b}\in F^*:\) \(a+(b+c)=(a+b)+d_{a,b}\cdot c.\) The conditions (F3), (F4) tell us, that each \(d^._{a,b}: F\to F\); \(x\to d_{a,b}\cdot x\) is an automorphism of the loop \((F,+)\). This observation led H. Wefelscheid to the notion of a K-loop \((F,+,d)\), which is defined by (F1) and (K) \(d:F\times F\to Aut(F,+)\); \((a,b)\to d_{a,b}\) is a map such that \(a+(b+c)=(a+b)+d_{a,b}(c)\). For a K-loop \((F,+,d)\) the function d has all the same properties (mentioned for instance in the paper by \textit{W. Kerby} and \textit{H. Wefelscheid} [J. Algebra 28, 319-325 (1974; Zbl 0276.16029)]) as in the case of a near- domain. While the question whether there are proper near-domains \((F,+,\cdot)\) (i.e. \((F,+)\) is not a group) has not been answered yet, the author presents here a very interesting class of examples of proper K-loops F, where the addition of F is defined as the relativistic velocity addition in modern physics: Leet (V,\({\mathbb{R}},\cdot)\) be a Euclidean vector space and for \(c\in {\mathbb{R}}\), \(c>0\) let \(V_ c:=\{{\mathfrak x}\in V|\| {\mathfrak x}\|:=\sqrt{{\mathfrak x}\cdot {\mathfrak x}}<c\}\). Then for each \({\mathfrak x}\in V_ c\), \(\gamma_{{\mathfrak x}}:=(1- ({\mathfrak x}/c)^ 2)^{-1/2}\) is a positive real number and \(V_ c\) is closed with respect to the binary operation \[ {\mathfrak x}\oplus {\mathfrak y}:=(1+({\mathfrak x}\cdot {\mathfrak y})/c^ 2)^{-1}[({\mathfrak x}+{\mathfrak y})- (1/c^ 2)(\gamma_{{\mathfrak x}}/1+\gamma_{{\mathfrak x}})({\mathfrak x}^ 2\cdot y-({\mathfrak x}\cdot {\mathfrak y})\cdot {\mathfrak x})]. \] The author shows that \((V_ c,\oplus,d)\) is a K-loop. The correction function d is here the Thomas rotation (introduced by L. H. Thomas 1926 in connection with the motion of the spinning electron) defined by \(d_{{\mathfrak a,b}}:=Id+\alpha ({\mathfrak a}\square {\mathfrak b}-{\mathfrak b}\square {\mathfrak a})+\beta ({\mathfrak a}\square {\mathfrak b}-{\mathfrak b}\square {\mathfrak a})^ 2\) where \({\mathfrak a}\square {\mathfrak b}: V\to V\); \({\mathfrak x}\to {\mathfrak a}({\mathfrak b}\cdot {\mathfrak x})\), \[ \alpha:=+(1/c^ 2)\gamma_{{\mathfrak a}}\cdot \gamma_{{\mathfrak b}}(1+\gamma_{{\mathfrak a}}+\gamma_{{\mathfrak b}}+\gamma_{{\mathfrak a}\oplus {\mathfrak b}})/(1+\gamma_{{\mathfrak a}})(1+\gamma_{{\mathfrak b}})(1+\gamma_{{\mathfrak a}\oplus {\mathfrak b}})\text{ and } \gamma:=(1/c^ 4)\gamma^ 2_{{\mathfrak a}}\gamma^ 2_{{\mathfrak b}}\cdot 1/(1+\gamma_{{\mathfrak a}})(1+\gamma_{{\mathfrak b}})(1+\gamma_{{\mathfrak a}\oplus {\mathfrak b}}). \] The Thomas rotation \(d_{{\mathfrak a},{\mathfrak b}}\) has the additional properties that \(d_{{\mathfrak a},{\mathfrak b}}\) is adjoint to \(d_{{\mathfrak b},{\mathfrak a}}\), and \(d_{{\mathfrak a},{\mathfrak b}}\) is an isometry of \((V,{\mathbb{R}},\| \|)\). For \({\mathfrak a},{\mathfrak b}\in V\) linearly independent \(d_{{\mathfrak a,b}}\) is a rotation of \((V,\| \|)\) with the axis \({\mathfrak a}^{\perp}\cap {\mathfrak b}^{\perp}\) and the angle \[ \epsilon:=\arccos(1-(1+2\gamma_{{\mathfrak a}}\gamma_{{\mathfrak b}}\gamma_{{\mathfrak a}\oplus {\mathfrak b}}-\gamma^ 2_{{\mathfrak a}}- \gamma^ 2_{{\mathfrak b}}-\gamma^ 2_{{\mathfrak a}\oplus {\mathfrak b}})/(1+\gamma_{{\mathfrak a}})(1+\gamma_{{\mathfrak b}})(1+\gamma_{{\mathfrak a}\oplus {\mathfrak b}})) \] and for \({\mathfrak a},{\mathfrak b}\) linearly dependent \(d_{{\mathfrak a},{\mathfrak b}}=id\), or if \(\Theta =\sphericalangle ({\mathfrak a},{\mathfrak b})\), then cos \(\epsilon\) \(=((k+\cos \Theta)^ 2-\sin^ 2\Theta)/((k+\cos \Theta)^ 2+\sin^ 2\Theta)\), if \(k>1\) and \(k^ 2=(\gamma_{{\mathfrak a}}+1)/(\gamma_{{\mathfrak a}}-1)\cdot (\gamma_{{\mathfrak b}}+1)/(\gamma_{{\mathfrak b}}-1).\) Following the definition of the correspondence between near-domains and sharply 2-transitive groups [the reviewer, op.cit.] one extends this procedure to K-loops: Let \((F,+,d)\) be a K-loop, \(\Phi \leq Aut(F,+)\) with \(d_{a,b}\in \Phi\) for all a,b\(\in F\) and for \(a\in F\) let \(a^+: F\to F\); \(x\to a+x\). Then the set \(G:=\{a^+\circ B|\) \(a\in F\), \(B\in \Phi \}\) of permutations of F forms a group since \(a^+\circ A\circ b^+\circ B=(a+Ab)^+\circ d_{a,Ab}\circ A\circ B\). The author calls this group \(G:=F{\mathcal Q}\Phi\) the quasidirect product between the K-loop F and the group \(\Phi\). For the special case \(V={\mathbb{R}}^ 3\), hence \(({\mathbb{R}}^ 3_ c,\oplus)\) we have \(d_{{\mathfrak a,b}}\in SO(3)\), so that we can form \(G={\mathbb{R}}^ 3{\mathcal Q}SO(3)\). Then G is the homogeneous, proper orthochronous Lorentz group of the special theory of relativity. Remark. \textit{G. Kist} introduced 1980 in his Habilitationsschrift ``Theorie der verallgemeinerten kinematischen Räume'' [cf. p. 3, 19, 20 of TUMM 8611, Technische Universität München. Mathematisches Institut, see also Result. Math. 12, 325-347 (1987; Zbl 0636.51012)]) the notion ``quasi-domain'' which is equivalent to the notion K-loop and formed the quasidirect product without using this name.