gyrogroup

A gyrogroup is an algebraic structure that generalizes groups by allowing controlled nonassociativity. It consists of a set G with a binary operation ⊕ and a family of automorphisms gyr[a,b] of (G, ⊕) that describe how far the operation deviates from associativity. The standard axioms include the existence of a neutral element e with e ⊕ a = a for all a in G, the existence of inverses (each a has a left inverse a' with a' ⊕ a = e), and two main laws: the gyroassociative law a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a,b](c) for all a,b,c in G, and the automorphism property gyr[a,b](x ⊕ y) = gyr[a,b](x) ⊕ gyr[a,b](y). In many formulations a gyrocommutative law a ⊕ b = gyr[a,b](b ⊕ a) is also included. These axioms ensure a controlled form of nonassociativity, with gyr[a,b] capturing the deviation.

The primary motivating example is Einstein velocity addition on the open unit ball of Euclidean space, where the operation u ⊕ v represents relativistic addition of velocities. The corresponding gyrations are Lorentz boosts, which act as rotations in the appropriate frame. When all gyrations are the identity map, a gyrogroup reduces to an ordinary group.

Gyrogroups underpin the theory of gyrovector spaces, which pair a gyrogroup operation with scalar multiplication to model hyperbolic geometry in a vector-space-like framework. They have applications in physics, particularly special relativity and hyperbolic geometry, as well as in algorithms that operate in hyperbolic spaces.