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

Following discussion here

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 25th 2019

• 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:

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

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

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeAug 26th 2019

$G_3$ 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).

• 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

• Kasper Andersen, Jesper Grodal, The classification of 2-compact groups, (arXiv:math/0611437)
• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeAug 26th 2019

Added details about the ring of invariants.

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

Jack Morava writes in to remark that the notation $G_3$ 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, $DI(0) = 1 = Aut_{\mathbb{R}}(\mathbb{R})$, we may complete the table as follows, and I have made that edit in the entry:

$n=$ 0 1 2 3 4
$DI(n)=$ 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 $G_3$ is from the automorphisms of the sedenions. Or, better yet, how far that thing that $G_3$ 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”…

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

Added references to homotopy colimit construction.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2019

Something on the relationship to the third Conway group.

• 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’ $B D I(4)$ which contains $Co_3$ and looks as though it should be some kind of twist of ‘skew-symmetric $3 \times 3$ 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 $Co_3$ link.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 27th 2019

Now if what Wilson means by “skew-symmetric $3 \times 3$-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 $G_3$ is “some kind of twist skew-symmetric $3 \times 3$-matrices over the octonions”. Does this come from $CO_3$ inside it?

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2019

I was wondering whether that idea of centrally extending $\mathbb{R}^{0|3}$ 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 $S H(3, \mathbb{O})$ of $3 \times 3$ 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 $3 \times 3$ octonion matrices would be the Lie algebra of the unitary $3 \times 3$ matrices, which in turn would act on the $3 \times 3$ 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 $G_3$?!

• 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 $C_3$ instead of $B_3$. “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 $U(3,\mathbb{O})$. 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.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeAug 28th 2019

I guess given the action of $SO(3)$ on imaginary quaternions (dim 3) and $G_2$ 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 $G_3$ 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. $G_3$ 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 $3 \times 3$-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 $G_2$ 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 $3 \times 3$ 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.

• 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 $B$ and $C$ 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 $Co_0$. There’s plenty of ’physics’ talk, e.g., Theo:

My motivation was the following. There is a holomorphic $N=1$ SCFT called $V^{f\natural}$ with automorphism group $Co_1$, first constructed by John Duncan. This $Co_1$ 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 $Co_1$ to its double cover $Co_0$. Second, there is still an $H^4(Co_0; Z)$ valued anomaly, which can be canceled by anomaly-inflow from a 3D Dijkgraaf—Witten model. Equivalently, to trivialize the anomaly requires pulling back from $Co_0$ 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 $H^4(Co_0; Z)$, namely the fractional Pontryagin class of the 24-dimensional representation.

• CommentRowNumber32.
• CommentAuthorDavid_Corfield
• CommentTimeSep 4th 2019

• Michael Aschbacher, Andrew Chermak, A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, (paper)
• CommentRowNumber33.
• CommentAuthorGuest
• CommentTimeJan 10th 2020
I think the claims about the Euler characteristic of G_3/Spin(7) of Dwyer-Wilkerson are false. The mod-2 Euler characteristic is 15, and the rational Euler characteristic is 7. The latter follows from the fact that 7 is the index of the Weyl group of Spin(7) in the Weyl group of G_3. Both claims are corroborated in
J. Aguadé, "The torsion index of a p-compact group", Proceedings of the AMS 138(11) , 2010, p. 4133.

- Tilman Bauer
• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeJan 10th 2020

Thanks. Made a brief note in the entry, here. Please feel invited to edit.

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeJan 10th 2020

Notbohm agrees with Aguadé, referring to a misprint in Dwyer-Wilkerson, so I’ve corrected and added this location.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJan 10th 2020
• (edited Jan 10th 2020)

Excellent, thanks!