The Split-Algebras and Non-compact 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.

40 pages, no figures, + 8 pages, discussions and references added, published version

References in corpus (6)

• Doubled Geometry and T-Folds
• Fuzzy Supersphere and Supermonopole
• A Hopf bundle over a quantum four-sphere from the symplectic group
• Split-Quaternionic Hopf Map, Quantum Hall Effect, and Twistor Theory
• Second Hopf map and supersymmetric mechanics with Yang monopole
• Hyperbolic Supersymmetric Quantum Hall Effect

Cited by in corpus (6)

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• Spin-entangled Squeezed State on a Bloch Four-hyperboloid
• BPS Black Hole Entropy and Attractors in Very Special Geometry. Cubic Forms, Gradient Maps and their Inversion