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nLab > Latest Changes: Moufang loop

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 14th 2013

    Created Moufang loop and some links. It would be good to update the proof that the tangent bundle of a Lie group is trivial to include the case of the tangent bundle of a smooth Moufang loop.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2013

    Under quasigroup I proved that the tangent bundle of a smooth quasigroup is trivial. By the way, who is that quote in quasigroup due to?

    Thanks for writing Moufang loop, David. A few questions/comments on examples: since the first example already says that a group is a Moufang loop, I’m not sure you need to say in the next example that the subgroup of quaternion units in the octonions forms a Moufang loop (and also, since there are many ways in which the quaternions sit inside the octonions, I wouldn’t say “the” subgroup of quaternions, if you see what I mean). If it were me, I’d leave that out, and replace it with the observation that more generally, the units in an alternative ring/algebra form a Moufang loop.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2013
    • (edited Apr 15th 2013)

    That was a typo, I meant to say unit octonions. I’ve fixed it. Actually I should probably say octonions “of unit length”, since I don’t mean units in the algebraic sense.

    Thanks for the proof at quasigroup.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2013

    Made some more edits, including the observation that Moufang loops are algebras for a Lawvere theory.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2013
    • (edited Apr 15th 2013)

    The most interesting thing I have learned from writing this, is the loop that Conway used to construct the Monster. In this paper loops are constructed using cocycles on vector spaces over 𝔽2 valued in 𝔽2, where the vector spaces are given as subspaces of 𝔽X2=P(X) for finite sets X (see definition 6 to proposition 9).

    I’m thinking that there could be a finite 2-group or categorified gadget floating around.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2021

    cross-linked with loop (algebra)

    diff, v14, current

  7. Fix one of the Moufang identities to match parentheses

    Tej Qu Nair

    diff, v15, current

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

    Anonymouse

    diff, v16, current

1 to 8 of 8

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