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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.
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.
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.
Made some more edits, including the observation that Moufang loops are algebras for a Lawvere theory.
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.
cross-linked with loop (algebra)
changed higher algebra - contents to algebra - contents in context sidebar
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