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Added:
$G_3$ 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).
Thanks. I have moved this from the Idea-section to the Properties-section, and added pointer to p-compact group.
Added reference
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]
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”…
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’.
Thanks for the pointer! Have added that to the entry, too (here)
Should make it “Robert A. Wilson”, but you are editing now…
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?
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)
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.
I have recorded this in the entry, here
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 $G_3$?!
[ … ]
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.”
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,
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.
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?
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?
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.
Thanks. What’s your references Benson98 ?
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.
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