Gyrovector space

A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups. Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts – "boosts" are aspects of relative velocities, and should not be conflated with "translations"). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity. Gyrogroups are weakly associative group-like structures. Ungar proposed the term gyrogroup for what he called a gyrocommutative-gyrogroup, with the term gyrogroup being reserved for the non-gyrocommutative case, in analogy with groups vs. abelian groups. Gyrogroups are a type of Bol loop. Gyrocommutative gyrogroups are equivalent to K-loops although defined differently. The terms Bruck loop and dyadic symset are also in use. A gyrogroup (G, ) consists of an underlying set G and a binary operation satisfying the following axioms: In G there is at least one element 0 called a left identity with 0 a = a for all a in G. For each a in G there is an element a in G called a left inverse of a with (a) a = 0. For any a, b, c in G there exists a unique element gyr[a,b]c in G such that the binary operation obeys the left gyroassociative law: a (b c) = (a b) gyr[a,b]c The map gyr[a,b]: G → G given by c ↦ gyr[a,b]c is an automorphism of the magma (G, ) – that is, gyr[a,b] is a member of Aut(G, ) and the automorphism gyr[a,b] of G is called the gyroautomorphism of G generated by a, b in G. The operation gyr: G × G → Aut(G, ) is called the gyrator of G. The gyroautomorphism gyr[a,b] has the left loop property gyr[a,b] = gyr[a b,b] The first pair of axioms are like the group axioms.