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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 $\mathbb{F}_2$ valued in $\mathbb{F}_2$, where the vector spaces are given as subspaces of $\mathbb{F}_2^X = P(X)$ 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)
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