advanced_tools:hopf_bundle
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advanced_tools:hopf_bundle [2017/12/20 10:44] jakobadmin [Examples] |
advanced_tools:hopf_bundle [2017/12/20 10:45] jakobadmin [Examples] |
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| Quaternions $\mathbb{H}$ are used to define the Hopf map $S^7 \to S^4$. This Hopf map describes a single instanton. | Quaternions $\mathbb{H}$ are used to define the Hopf map $S^7 \to S^4$. This Hopf map describes a single instanton. | ||
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| Octonions $\mathbb{O}$ are used to define the Hopf map $S^{15} \to S^8$. **Currently there is no physics application known of this map!** This map is different from the other two, because the fibre $S^7$ (the unit octonions) is not really a group. The reason for this is that octonions aren't associative. | Octonions $\mathbb{O}$ are used to define the Hopf map $S^{15} \to S^8$. **Currently there is no physics application known of this map!** This map is different from the other two, because the fibre $S^7$ (the unit octonions) is not really a group. The reason for this is that octonions aren't associative. | ||
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| (The classification of the Hopf bundles as listed here is surprisingly similar to the [[http://jakobschwichtenberg.com/classification-of-simple-lie-groups/|classification of all simple Lie groups]]. Each Hurwitz algbra corresponds to one family of simple groups. The octonions play a special role, because they correspond to the exceptional family, which has only a finite number of members.) | (The classification of the Hopf bundles as listed here is surprisingly similar to the [[http://jakobschwichtenberg.com/classification-of-simple-lie-groups/|classification of all simple Lie groups]]. Each Hurwitz algbra corresponds to one family of simple groups. The octonions play a special role, because they correspond to the exceptional family, which has only a finite number of members.) | ||
| - | For a summary how $S^7$ could be used in physics see http://math.ucr.edu/home/baez/week141.html | ||
| <tabbox FAQ> | <tabbox FAQ> | ||
advanced_tools/hopf_bundle.txt ยท Last modified: 2018/05/03 11:07 by jakobadmin
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