Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 𝔽 2\mathbb{F}_2 valued in 𝔽 2\mathbb{F}_2, where the vector spaces are given as subspaces of 𝔽 2 X=P(X)\mathbb{F}_2^X = P(X) for finite sets XX (see definition 6 to proposition 9).

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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)