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nLab > Latest Changes: octonion

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016
    • (edited Jan 20th 2016)

    Added the definition of “basic triples” of octonions, and the statement that they form a torsor over Aut(𝕆)=G 2Aut(\mathbb{O}) = G_2.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2016

    added the actual definition to octonions

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2018

    added statement and proof (here) that the product of all the seven imaginary quaternions with each other is ±1\pm 1.

    diff, v16, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2018
    • (edited Apr 23rd 2018)

    Added a statement (here) concerning projecting out ℍ\mathbb{H} from 𝕆\mathbb{O}.

    diff, v17, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    added statement and proof (here) that the octonions are alternative

    diff, v18, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    made explicit the definition of real and imaginary octonions here

    diff, v18, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018

    stated the Clifford action of the imaginary octonions induced by left multiplication (here)

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2018
    • (edited Apr 24th 2018)

    added statement and Clifford-theoretic proof (here) that the consecutive left product by all the seven imaginary generators acts as the identity

    diff, v19, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2018

    Determined the remaining sign ±1\pm 1 in that prop:

    L e 7L e 6L e 5L e 4L e 3L e 2L e 1=+1 L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} = + 1

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2020
    • (edited Apr 17th 2020)

    I would like to replace the Fano plane diagram for the octonion multiplication (here) with one whose labels are more well-adapted to applications: The labels e 1e_1, e 2e_2, e 3e_3 should be on one line, and maybe best on the inner circle, so that one can readily identify them with the generators of a copy of the quaternions inside the octonions.

    Checking, I see that on Wikipedia they had just the same idea for re-labeling: here.

    But there they also reversed the direction of the straight inner lines. Hm, is that irrelevant up to isomorphism? Or is that a mistake?

    [ Never mind. I see it’s consistent with Cayley-Dickson. ]

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2020
    • (edited Apr 18th 2020)

    Okay, I have made an improved graphics of the “octonion multiplication table”, now well-adapted to inclusion of the quaternions, with the generators labeled according to their Dickson-double incarnation: here.

    Now including it into the entry…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    I have added pointer to the original references

    • Arthur Cayley, On certain results relating to quaternions, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 (doi:10.1080/14786444508562684)

    • Leonard Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem, Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)

    Then I adjusted the first sentences in the Idea-section for clarity,and then expanded the Idea-section to provide some minimum of perspective.

    diff, v28, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    added pointer also to

    and will add this to various related entries, too.

    diff, v28, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    have re-typed the previous example for some basic octonion algebra, now adjusted to the new labeling of generators:

    e 4(e 5(e 6(e 7x))) =ℓ((iℓ)((jℓ)((kℓ)x))) =ℓ((iℓ)((jℓ)((kx¯)ℓ))) =ℓ((iℓ)((xk)j)) =ℓ((i(j(kx¯))ℓ) =((xk)j)i ={−x if x∈ℍℓ↪𝕆 −x if x∈ℍℓ↪𝕆 \begin{aligned} \mathrm{e}_4 \Big( \mathrm{e}_5 \big( \mathrm{e}_6 (\mathrm{e}_7 x) \big) \Big) & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \ell) x \big) \Big) \bigg) \\ & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \overline{x}) \ell \big) \Big) \bigg) \\ & = \ell \Big( (i \ell) \big( (x k) j \big) \Big) \\ & = \ell \bigg( \Big( i \big( j (k \overline{x} \big) \Big) \ell \bigg) \\ & = \big( (x k) j \big) i \\ & = \left\{ \begin{array}{ccc} \phantom{-}\, x & \text{if} & x \in \mathbb{H}\phantom{\ell} \hookrightarrow \mathbb{O} \\ - x & \text{if} & x \in \mathbb{H}\ell \hookrightarrow \mathbb{O} \end{array} \right. \end{aligned}

    diff, v29, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeMay 14th 2024
    • (edited May 14th 2024)

    The definition of ’basic triple’ is wrong: if we take i, j, k in a quaternion subalgebra of the octonions it count as a basic triple according to this definition, but that’s not right. One needs to add that each one anticommutes with the product of the other two. I’ll fix this.

  16. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v37, current

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