From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups - Abraham A. Ungar (Mathematics Interdisciplinary Research, 2016)
‎The Lorentz transformation of order (m=1,n)‎, ‎n∈N‎, ‎is the well-known ‎Lorentz transformation of special relativity theory‎. ‎It is a transformation of time-space coordinates of the ‎pseudo-Euclidean space Rm=1,n of one time dimension and n space dimensions (n=3 in physical applications)‎. ‎A Lorentz transformation without rotations is called a boost. ‎Commonly‎, ‎the special relativistic boost is ‎parametrized by a relativistically admissible velocity parameter v‎, ‎v ∈ Rcn‎, ‎whose domain is the c-ball Rcn of all ‎relativistically admissible velocities‎, ‎Rcn={v ∈ Rn‎: ||v||<c}‎, ‎where the ambient space Rn is the ‎Euclidean n-space‎, ‎and c>0 is an arbitrarily fixed ‎positive constant that represents the vacuum speed of light‎. ‎The study of the Lorentz transformation composition law in terms of ‎parameter composition reveals that the group structure of the ‎Lorentz transformation of order (m=1,n) induces a gyrogroup and ‎a gyrovector space structure that regulate ‎the parameter space Rcn‎‎. ‎The gyrogroup and gyrovector space structure ‎of the ball Rcn‎‎, ‎in turn‎, ‎form the algebraic setting for the Beltrami-Klein ball model ‎of hyperbolic geometry‎, ‎which underlies the ball Rcn‎‎. ‎The aim of this article is to extend the study of the ‎Lorentz transformation of order (m,n) from m=1 and n≥1 to ‎all m,n∈N‎, ‎obtaining algebraic structures called ‎a bi-gyrogroup and a  bi-gyrovector space‎. ‎A bi-gyrogroup is ‎a gyrogroup each gyration of which is a pair of ‎a left gyration and a right gyration‎. ‎A bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup ‎that admits a scalar multiplication‎.