Mathematics Interdisciplinary Research
Volume:1 Issue: 1, Winter 2016

This autobiography presents the scientific living of Abraham Ungar and his role in Gyrogroups and Gyrovector spaces.‎

The only justification for the Einstein velocity addition law ‎appeared to be its empirical adequacy‎, ‎so that the ‎intrinsic beauty and harmony in Einstein addition remained for a long time ‎a mystery to be conquered‎. ‎Accordingly‎, ‎the aim of this expository article is to present ‎(i) the Einstein relativistic vector addition‎, ‎(ii) the resulting Einstein scalar multiplication‎, ‎(iii) the Einstein relativistic mass‎, ‎and ‎(iv) the Einstein relativistic kinetic energy‎, ‎along with remarkable analogies with classical results in ‎groups and vector spaces that these ‎Einstein concepts capture in gyrogroups and gyrovector spaces‎. ‎Making the unfamiliar familiar‎, ‎these analogies uncover the ‎intrinsic beauty and harmony in the underlying ‎Einstein velocity addition law of relativistically admissible velocities‎, ‎as well as its interdisciplinarity‎.

‎A gyrogroup is a nonassociative group-like structure modelled on the ‎space of relativistically admissible velocities with a binary ‎operation given by Einstein's velocity addition law‎. ‎In this ‎article‎, ‎we present a few of groups sitting inside a gyrogroup G‎, ‎including the commutator subgyrogroup‎, ‎the left nucleus‎, ‎and the ‎radical of G‎. ‎The normal closure of the commutator subgyrogroup‎, ‎the left nucleus‎, ‎and the radical of G are in particular normal ‎subgroups of G‎. ‎We then give a criterion to determine when a ‎subgyrogroup H of a finite gyrogroup G‎, ‎where the index ‎$[Gcolon H]$ is the smallest prime dividing |G|‎, ‎is normal in G‎.

‎Einstein‎, ‎M"{o}bius‎, ‎and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time $bkR^{n,1}.$ Using the gyrolanguage we study their gyroharmonic analysis‎. ‎Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them‎. ‎Our study focus on the translation and convolution operators‎, ‎eigenfunctions of the Laplace-Beltrami operator‎, ‎Poisson transform‎, ‎Fourier-Helgason transform‎, ‎its inverse‎, ‎and Plancherel's Theorem‎. ‎We show that in the limit of large $t,$ $t rightarrow‎ +‎infty,$ the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on ${mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis‎.

‎The decomposition $Gamma=BH$ of a group $Gamma$ into a subset B ‎and a subgroup $H$ of $Gamma$ induces‎, ‎under general conditions‎, ‎a ‎group-like structure for B‎, ‎known as a gyrogroup‎. ‎The famous‎ concrete realization of a gyrogroup‎, ‎which motivated the emergence ‎of gyrogroups into the mainstream‎, ‎is the space of all ‎relativistically admissible velocities along with a binary ‎mbox{operation} given by the Einstein velocity addition law of ‎special relativity theory‎. ‎The latter leads to the Lorentz ‎transformation group $so{1,n}$‎, ‎$ninN$‎, ‎in pseudo-Euclidean ‎spaces of signature $(1‎, ‎n)$‎. ‎The study in this article is motivated ‎by generalized Lorentz groups $so{m‎, ‎n}$‎, ‎$m‎, ‎ninN$‎, ‎in ‎pseudo-Euclidean spaces of signature $(m‎, ‎n)$‎. ‎Accordingly‎, ‎this ‎article explores the bi-decomposition $Gamma = H_LBH_R$ of a group ‎$Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of ‎$Gamma$‎, ‎along with the novel bi-gyrogroup structure of $B$ induced ‎by the bi-decomposition of $Gamma$‎. ‎As an example‎, ‎we show by ‎methods of Clifford mbox{algebras} that the quotient group of the ‎spin group $spin{m‎, ‎n}$ possesses the bi-decomposition structure‎.

‎In this paper‎, ‎we consider a generalization of the real normed spaces and give some examples‎.

‎In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces‎, ‎which are the Einstein and M"{o}bius gyrovector spaces‎. ‎We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its interesting applications on the set of invertible density matrices‎. ‎Finally we give an example of the gyrovector space on the unit ball of Hermitian matrices‎.

‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geometry‎. ‎In this paper‎, ‎we directly use the isomorphism properties of gyrovector spaces to recover the Chen's addition and then Chen model of hyperbolic geometry‎. ‎We show that this model is an extension of the Poincar'e model of hyperbolic geometry‎. ‎For our purpose we consider ‎the Poincar'e plane model of hyperbolic geometry inside the complex open unit disc $mathbb{D}$‎. ‎Also we prove that this model is isomorphic to the Poincar'e model and then to other models of hyperbolic geometry‎. ‎Finally‎, ‎by gyrovector space approach we verify some properties of this model in details in full analogue with Euclidean geometry‎.

‎This paper extends the scope of algebraic computation based on a non standard $times$ to ‎the more basic case of a non standard $+$‎, ‎where standard means associative ‎and commutative‎. ‎Two physically meaningful examples of a non standard $+$ are ‎provided by the observation of motion in Special Relativity‎, ‎from either ‎outside (3D) or inside (2D or more)‎, ‎We revisit the ``gyro''-theory of Ungar to present ‎the multifaceted information processing which is created by a metric cloth $W$‎, ‎a relating computational construct framed in a normed vector space $V$‎, ‎and based ‎on a non standard addition denoted $pluscirc$ whose commutativity and associativity ‎are ruled (woven) by a relator‎, ‎that is a map which assigns to each pair of admissible vectors ‎in $V$ an automorphism in $Aut W$‎. ‎Special attention is given to the case where the relator is ‎directional‎.

‎The Lorentz transformation of order $(m=1,n)$‎, ‎$ninNb$‎, ‎is the well-known ‎Lorentz transformation of special relativity theory‎. ‎It is a transformation of time-space coordinates of the ‎pseudo-Euclidean 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, ‎$Rcn={vbinRn:|vb|<c}$‎, ‎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 $nge1$ to ‎all $m,ninNb$‎, ‎obtaining algebraic structures called ‎a {it bi-gyrogroup} and a {it 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.