On quadratic and nonquadratic forms: Application to nonbijective R^{2m} -> R^{2m-n} transformations

TL;DR

This paper explores Hurwitz transformations related to Cayley-Dickson algebras of dimensions 2, 4, and 8, analyzing their role in generating quadratic and nonquadratic forms with applications in number theory and dynamical systems.

Contribution

It provides a detailed investigation of Hurwitz transformations for specific Cayley-Dickson algebras, highlighting their geometric and Lie-algebraic properties and applications.

Findings

Identification of Hurwitz transformations leading to quadratic forms

Analysis of transformations in dimensions 2, 4, and 8

Brief applications to number theory and dynamical systems

Abstract

Hurwitz transformations are defined as specific automorphisms of a Cayley-Dickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to quadratic forms are discussed from geometrical and Lie-algebraic points of view. Applications to number theory and dynamical systems are briefly examined.