The Split-Algebras and Non-compact Hopf Maps

TL;DR

This paper introduces noncompact Hopf maps constructed from split algebras, exploring their topological structures and gauge field interpretations, expanding the understanding of hyperboloid bundles in mathematical physics.

Contribution

It develops a noncompact version of Hopf maps using split algebras, providing new topological insights and gauge field realizations.

Findings

Constructed noncompact Hopf maps using split algebras.

Analyzed topological structures of hyperboloid bundles.

Identified canonical connections as noncompact gauge fields.

Abstract

We develop a noncompact version of the Hopf maps based on the split algebras. The split algebras consist of three species: split-complex numbers, split quaternions, and split octonions. They correspond to three noncompact Hopf maps that represent topological maps between hyperboloids in different dimensions with hyperboloid bundle. We realize such noncompact Hopf maps in two ways: one is to utilize the split-imaginary unit, and the other is to utilize the ordinary imaginary unit. Topological structures of the hyperboloid bundles are explored, and the canonical connections are naturally regarded as noncompact gauge field of monopoles.