Abstract :
The Lorentz transformation of order $(m=1,n)$, $ninNb$, is the wellknown Lorentz transformation of special relativity theory. It is a transformation of timespace coordinates of the pseudoEuclidean space $Rb^{m=1,n}$ of one time dimension and $n$ space dimensions ($n=3$ in physical applications). A Lorentz transformation without rotations is called a {it boost}. Commonly, the special relativistic boost is parametrized by a relativistically admissible velocity parameter $vb$, $vbinRcn$, whose domain is the $c$ball $Rcn$ of all relativistically admissible velocities, $Rcn={vbinRn:|vb| lt;c}$, where the ambient space $Rn$ is the Euclidean $n$space, and $c gt;0$ 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 BeltramiKlein 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 $nge1$ to all $m,ninNb$, obtaining algebraic structures called a {it bigyrogroup} and a {it bigyrovector space}. A bigyrogroup is a gyrogroup each gyration of which is a pair of a left gyration and a right gyration. A bigyrovector space is constructed from a bigyrocommutative bigyrogroup that admits a scalar multiplication.
