ar X iv : 0 90 5 . 27 92 v 1 [ m at h - ph ] 1 8 M ay 2 00 9 The Split -

We develop a non-compact 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 non-compact Hopf maps that represent topological mapping between higher dimensional hyperboloids in different dimensions with hyperboloid-bundles. We realize such non-compact Hopf maps in two ways: one is to utilize the split-imaginary unit, and the other is to utilize the ordinary imaginary unit. The connections of the hyperboloid-bundles in the non-compact Hopf maps are evaluated, and they are naturally regarded as non-compact gauge fields of monopoles.