Implement the octonions #35630
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Implement the octonions #35630
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They do not seem to appear in the documentation. For instance under the setting "non-associative algebras". |
tscrim
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I thought I had fixed that. Well, now it is done. (With a force push since I had to rebase it anyways...) |
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There are several typos "octonian". |
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another typo : "quaterions" |
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Good catch(es). Thank you. |
mkoeppe
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May 16, 2023
| non-commutative unital 8-dimensional `R`-algebra that is a deformation | ||
| of the usual octonions, which are when `a = b = c = -1`. The octonions | ||
| were originally constructed by Graves and independently discovered by | ||
| Cayley (who due to first publishing them, they are sometimes called |
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"who due to ..., they are" does not parse
mkoeppe
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May 16, 2023
| r""" | ||
| Test that ``self`` is an Hurwitz algebra. | ||
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| An algebra `A` is *Hurwitz* if there exists a quadratic form `N` |
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nondegenerate? Otherwise this is vacuous
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Thank you! |
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Documentation preview for this PR (built with commit 4370091) is ready! 🎉 |
vbraun
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Jun 21, 2023
<!-- Please provide a concise, informative and self-explanatory title. --> <!-- Don't put issue numbers in the title. Put it in the Description below. --> <!-- For example, instead of "Fixes #12345", use "Add a new method to multiply two integers" --> ### 📚 Description <!-- Describe your changes here in detail. --> <!-- Why is this change required? What problem does it solve? --> <!-- If this PR resolves an open issue, please link to it here. For example "Fixes #12345". --> <!-- If your change requires a documentation PR, please link it appropriately. --> "The" exceptional Jordan algebra is a 27 dimensional algebra defined as the $3 \times 3$ self-adjoint matrices over the octonions. This is an important Jordan algebra as it is used to construct the simple Lie group/algebra of type $F_4$. Fixes #32940. ### 📝 Checklist <!-- Put an `x` in all the boxes that apply. It should be `[x]` not `[x ]`. --> - [x] The title is concise, informative, and self-explanatory. - [x] The description explains in detail what this PR is about. - [x] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies - #35630 used in the construction <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> URL: #35629 Reported by: Travis Scrimshaw Reviewer(s): Frédéric Chapoton, Matthias Köppe, Travis Scrimshaw
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📚 Description
The octonions are a nonassociative normed division algebra, one of only 4 over the reals, and used to create the 7 dimensional cross product. We implement a slight generalization with 3 parameters over an arbitrary commutative ring of characteristic not equal to 2, which allows us to also define the split-octonions as another special case.
Fixes #32940.
📝 Checklist
⌛ Dependencies