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nLab > Latest Changes: cross product

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeSep 17th 2018

    Fix k=0k = 0 case of vector-valued cross products (one for each unit vector).

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2018
    • (edited Sep 17th 2018)

    added this paragraph:

    By one of the deeper strands of mathematics, this classification of something as innocent-looking as cross-products is closely related not just to the existence of normed division algebras over the real numbers, but also to all of the following: parallelizable n-spheres, the existence of real spin representations, the homotopy groups of spheres of Hopf invariant one; see there

    diff, v15, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2018

    We do have Hurwitz theorem. Made the link work

    diff, v15, current

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeAug 23rd 2020
    • (edited Aug 23rd 2020)

    Relationship to Jordan–Lie algebras. (There should be a common generalization.)

    diff, v16, current

  5. Fixed the statement “combining the dot product with any vector-valued cross product produces a scalar-valued cross product of 1 higher arity”, which doesn’t hold in general: trying this with the 7-dimensional vector-valued binary cross product (using e.g. the table in Wikipedia), we have 0=(e 1×e 2)⋅e 4≠|e 1||e 2||e 4|=10=(e_1 \times e_2)\cdot e_4 \neq |e_1| |e_2| |e_4| = 1, contradicting Property 3.

    Anonymous

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeDec 20th 2020
    • (edited Dec 20th 2020)

    Good catch, Anonymous!

    In other news, I've added a bunch of combinatorics related to how many examples of cross products are compatible with bases; I'm still a little unclear about how the ternary cross products in 88 dimensions work, so I couldn't count those, but I figured out all of the rest. Also, I fixed the convention on the co-unary cross products to match how the scalar-valued curl works in 22 dimensions; hopefully that doesn't mess anything up on any other page.

    diff, v20, current

1 to 6 of 6

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