Factorizations of Möbius gyrogroups.
The Möbius addition on the unit ball \(\mathbf B_1\) in a real Hilbert space \(H\) is a famous example of a gyrogroup which arose from the addition of velocities in the theory of relativity. A. Ungar has introduced and studied this class of loops which have appeared in various other areas of mathematics. The author of the paper under review describes most of the other instances where these loops play a role in Geometry and Algebra. In his work he follows the definitions and notations of Ungar. Considering on \(H\) the quadratic form \(Q(v)=\langle v\mid v\rangle\) the author uses the Clifford algebra \(\text{Cl}[H,Q]\). The canonical embedding of \(H\) in \(\text{Cl}[H,Q]\) defines the so called geometric multiplication of vectors in \(H\) which is used to give a neat form of the operation of the gyrogroup \(\mathbf B_1\). For a vector \(\omega\) on the unit sphere of \(H\) put \(L_\omega=\{t\omega\mid -1\leq\omega\leq 1\}\) and \(D_\omega=\omega^\perp\cap\mathbf B_1\). Then \(L_\omega\) and \(D_\omega\) are gyrosubgroups of \(\mathbf B_1\) and \(L_\omega\) is even a subgroup of \(\mathbf B_1\). The author shows (Theorem 2) that for a given \(\omega\) every element \(c\in\mathbf B_1\) has unique decompositions \(c=d_1l_1=l_2d_2\) with \(l_1,l_2\in L_\omega\), \(d_1,d_2\in D_\omega\). In the sequel it is shown that in \(\mathbf B_1\) the decompositions \(\mathbf B_1=D_\omega L_\omega=L_\omega D_\omega\) define coset partitions \(\{bD_\omega\mid b\in L_\omega\}\), \(\{D_\omega b\mid b\in L_\omega\}\), \(\{aL_\omega\mid a\in D_\omega\}\), \(\{D_\omega a\mid a\in L_\omega\}\). In the paper the members of these partitions are computed in coordinates. Furthermore, dualities between them are described.
scientific article; zbMATH DE number 5593785