分裂代数与非紧Hopf映射
The Split-Algebras and Non-compact Hopf Maps
摘要：
我们开发了一个非紧版本的Hopf映射的基础上分裂代数。分裂代数由三个种类组成：分裂复数、分裂四元数和分裂八元数。它们对应于三个非紧的Hopf映射，这些映射用双曲面丛表示不同维双曲面之间的拓扑映射。我们用两种方法来实现这种非紧的Hopf映射：一种是利用分裂虚单位，另一种是利用普通虚单位。讨论了双曲面丛的拓扑结构，并将正则联络自然地看作非紧的单极规范场。
英文摘要：
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.
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