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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2019

    for completeness, to go with the other entries in coset space structure on n-spheres – table

    v1, current

  1. Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “Spin(7)/G2 is the 7-sphere” from the top and added one for “Spin(7)/G2 is the 7-sphere” at the bottom of the page.)

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorperezl.alonso
    • CommentTimeSep 19th 2024

    Here’s something I find puzzling. In Stasheff 63, it is shown that even though the seven-sphere S 7S^7 is a H-space, it does not have a homotopy-associative product. The binary operation that makes S 7S^7 a H-space is essentially the product of unit octonions. Now, in Albuquerque & Majid 98, it is explained that the octonions are essentially obtained by starting with the real group algebra [ 2× 2× 2]\mathbb{R}[\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 ] and twisting the multiplication by some 2-cochain. Since this cochain is not a cocycle, the resulting multiplication is not associative, but the nontrivial associator will satisfy a 3-cocycle condition. This is essentially what leads to the observation in p.9 in Baez 01 that one has a (braided) fusion category of 2 3\mathbb{Z}_2 ^3-graded vector spaces with nontrivial associator. So based on this I thought S 7S^7 would admit a homotopy associative multiplication, but this is not the case. Seems something is lost here. Is this suggesting that it is the stack Spin(7)//G 2Spin(7)//G_2 that is a (braided) Lie 2-group?