The split algebras and noncompact Hopf maps

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.