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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2019
    • (edited Aug 25th 2019)

    Following discussion here

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2019

    added a tad more

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2019
    • (edited Aug 25th 2019)

    made explicit that the first three in the sequence are automorphism groups of real normed division algebras:

    n=n= 1 2 3 4
    DI(n)=DI(n)= Z/2 SO(3) G2 G3
    = Aut(C) = Aut(H) = Aut(O)

    This suggests that DI(n)DI(n) for n=0n = 0 makes sense and is the point. Is it?

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 26th 2019

    Added:

    G 3G_3 is the only exotic 2-group, or, in other words, the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group (Bendersky-Davis 07, p. 1).

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2019

    Thanks. I have moved this from the Idea-section to the Properties-section, and added pointer to p-compact group.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 26th 2019

    Added reference

    • Kasper Andersen, Jesper Grodal, The classification of 2-compact groups, (arXiv:math/0611437)

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 26th 2019

    Added details about the ring of invariants.

    diff, v7, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2019
    • (edited Aug 26th 2019)

    Jack Morava writes in to remark that the notation G 3G_3 should not be attributed to him, but to

    section 2.4 in

    Jesper M\oller, Homotopy Lie groups, Bulletin of the AMS 32 (1995) 413 -428

    I can’t edit right now, maybe later

    [edit: have added it to the entry now]

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2019
    • (edited Aug 26th 2019)

    Since, if I see correctly, DI(0)=1=Aut ()DI(0) = 1 = Aut_{\mathbb{R}}(\mathbb{R}), we may complete the table as follows, and I have made that edit in the entry:

    n=n= 0 1 2 3 4
    DI(n)=DI(n)= 1 Z/2 SO(3) G2 G3
    = Aut(R) = Aut(C) = Aut(H) = Aut(O)

    So it would be really interesting now to see how far G 3G_3 is from the automorphisms of the sedenions. Or, better yet, how far that thing that G 3G_3 is the actual automorphisms of is from the sedenions. Maybe the sedenions are a dead end, and nature is telling us to look for the “homotopy octonions”…

    diff, v8, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019
    • (edited Aug 27th 2019)

    Added references to homotopy colimit construction.

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    Something on the relationship to the third Conway group.

    diff, v11, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    Interesting. At the end of his talk ’A new approach to the Leech lattice’, Robert Wilson writes:

    Perhaps it will explain the ‘2-local group’ BDI(4)B D I(4) which contains Co 3Co_3 and looks as though it should be some kind of twist of ‘skew-symmetric 3×33 \times 3 matrices over octonions’.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Thanks for the pointer! Have added that to the entry, too (here)

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Should make it “Robert A. Wilson”, but you are editing now…

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    A little more on the Co 3Co_3 link.

    diff, v13, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Now if what Wilson means by “skew-symmetric 3×33 \times 3-matrices over the octonions” connects to the Albert algebra, that would be something, and a rough physics picture would immediately spring to mind.

    So where does he get this from, that G 3G_3 is “some kind of twist skew-symmetric 3×33 \times 3-matrices over the octonions”. Does this come from CO 3CO_3 inside it?

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    I was wondering whether that idea of centrally extending 0|3\mathbb{R}^{0|3} might be relevant.

    Not sure how Wilson’s reasoning has gone. Elsewhere there is

    As the Dwyer–Wilkerson 2-compact group is, in some sense, a 45-dimensional object, it is a plausible conjecture that it might have some connection to the 45-dimensional algebra SH(3,𝕆)S H(3, \mathbb{O}) of 3×33 \times 3 skew hermitian matrices over the octonions with bracket multiplication). (p. 175 of Conjectures on finite and p-local groups)

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Ah, interesting, thanks. So for one I learn that Wilson’s “skew-symmetric” did not refer to “hermitian” (as in the Albert algebra), but to “skew-hermitian”. But it’s still curious how close this is now to the Albert algebra.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    I have recorded this in the entry, here

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Hm, now there is an obvious relation, isn’t there:

    If it were not for non-associativity of the octonions, the skew-hermitian 3×33 \times 3 octonion matrices would be the Lie algebra of the unitary 3×33 \times 3 matrices, which in turn would act on the 3×33 \times 3 hermitian matrices in the Albert algebra by automorphisms.

    Only that non-associativity prevents these unitary matrices from actually forming a group… but so maybe they form the \infty-group G 3G_3?!

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019
    • (edited Aug 27th 2019)

    [ … ]

    • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    If you can see the bottom of this page from Google Books, Benson is wondering about a 45-dimensional algebra of skew-hermitian matrices over a 2-adic version of the octonions, and points to a problem with a 21-dim subalgebra being C 3C_3 instead of B 3B_3. “But there may be some twisted version of this which works.”

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2019

    Thanks! Unfortunatly, GoogleBooks decides to hide this from me. I’ll see if I get a copy elsewhere.

    Just started to look around if anyone had made a proposal for U(3,𝕆)U(3,\mathbb{O}). There is an MO-question here, but no real reply,

    • CommentRowNumber24.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2019

    Added tentative claim of a 15-dim homotopy representation.

    diff, v16, current

    • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 28th 2019

    I guess given the action of SO(3)SO(3) on imaginary quaternions (dim 3) and G 2G_2 on imaginary octonions (dim 7), 15 might be expected.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2019

    Okay, if that’s the pattern, then the number 15 here should be identified with the dimension of the space of imaginary sedenions, inside the 16-dimensional space of all sedenions.

    That goes against the grain of having G 3G_3 be the automorphisms of the Albert algebra, but okay. Or might there be a natural way in which the sedenions sit inside the Albert algebra?

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 29th 2019
    • (edited Aug 29th 2019)

    Putting things figuratively, I guess when a concept is put under this much pressure, we should only expect at best traces of features that were there before. G 3G_3 seems to be a vestige, hanging on by its finger tips in the 2-adic world. We might plausibly expect then a trace of the sedenions.

    Do we have a rationale for the 3×33 \times 3-aspect, beyond the numerology of 45? Benson in #22 speaks of a “tempting candidate”, but I haven’t seen an explanation of 3 dimensions, unless Wilson (#12) is pointing to this with his 3 octonionic dimensional construction of the Leech lattice.

    Does G 2G_2 have any such thing going on?

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2019

    Yeah, I don’t know if people have more than plain numerology 45 = 45 for the 3×33 \times 3 picture. I was initially assuming they must have, for otherwise it would not seem worth mentioning even. But if so, I haven’t seen it either.

    • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 30th 2019

    Added some rationale for the 3 dim proposal.

    diff, v17, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2019

    Thanks. What’s your references Benson98 ?

    • CommentRowNumber31.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 30th 2019

    It was there but had ) instead of }, so nothing appeared. Fixed now.

    This is the reference mentioned in #22, with the warning about BB and CC groups.

    It would be fun to work out how the sporadic finite simple groups show up. Over at the Café, John posted on Theo Johnson-Freyd and David Treumann’s calculation of the fourth cohomology of Co 0Co_0. There’s plenty of ’physics’ talk, e.g., Theo:

    My motivation was the following. There is a holomorphic N=1N=1 SCFT called V fV^{f\natural} with automorphism group Co 1Co_1, first constructed by John Duncan. This Co 1Co_1 action is anomalous, meaning that there is an obstruction to gauging it. The anomaly comes in two pieces. First, “the” Ramond sector of an SCFT is only well-defined up to isomorphism, and so symmetries of an SCFT act projectively on the Ramond sector, but to gauge a symmetry requires promoting that action to an honest action. This requires lifting from Co 1Co_1 to its double cover Co 0Co_0. Second, there is still an H 4(Co 0;Z)H^4(Co_0; Z) valued anomaly, which can be canceled by anomaly-inflow from a 3D Dijkgraaf—Witten model. Equivalently, to trivialize the anomaly requires pulling back from Co 0Co_0 to the appropriate 3-group.

    I claim that of the 24 3-groups, the one that trivializes this anomaly is specifically the one corresponding to our generator of H 4(Co 0;Z)H^4(Co_0; Z), namely the fractional Pontryagin class of the 24-dimensional representation.

    • CommentRowNumber32.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 4th 2019

    Added a reference:

    • Michael Aschbacher, Andrew Chermak, A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, (paper)

    diff, v21, current

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