Finite Lorentz Transformations, Automorphisms, and Division Algebras

TL;DR

This paper provides an explicit algebraic framework for finite Lorentz transformations in 10D Minkowski space using octonions, with implications for superstring theory and insights into automorphisms of division algebras.

Contribution

It introduces a novel algebraic parameterization of Lorentz transformations in 10D using octonions and describes automorphisms of quaternions and octonions via conjugation maps.

Findings

Explicit octonionic parameterization of Lorentz transformations.

Descriptions of automorphisms of quaternions and octonions.

Construction of special orthogonal groups using division algebra operations.

Abstract

We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.