On algebraic endomorphisms of the Einstein gyrogroup

TL;DR

This paper characterizes all continuous algebraic endomorphisms of the Einstein gyrogroup, showing they are derived from orthogonal linear transformations, thus revealing the structure of symmetries in relativistic velocity addition.

Contribution

It provides a complete description of continuous algebraic endomorphisms of the Einstein gyrogroup, linking them to orthogonal linear transformations.

Findings

All nonzero endomorphisms originate from orthogonal transformations.

The structure of endomorphisms is fully characterized.

The results clarify symmetry properties of the Einstein velocity addition.

Abstract

We describe the structure of all continuous algebraic endomorphisms of the open unit ball $\mathbf{B}$ of $\mathbb{R}^3$ equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on $\mathbb{R}^3$.