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On the AlgTop mailing list the other day, Jack Morava suggested/requested that somebody collects some information on…
...the two-local H-space G_3 (still waiting to be discovered by physicists)...
Perhaps this remark is better suited for MathOverflow or something similar, but I just realized that there is no Wikipedia page for G_3 (a.k.a. DI(4)).
In view of the importance of Lie theory in mathematics in general, I think it certainly deserves one, and I hope the experts will take up the challenge.
In kind reaction to me asking him for more background on what this is about, Jack writes this, which I trust I may reproduce here:
Dear Urs,
I’ve attached the original (1993) paper - it’s about a 2-complete H-space which looks (at 2) like the next in line of a series of (two) finite loop-spaces, starting with G2, along with a more recent paper which I should have known about, but didn’t until I started this response.
The analog of the Weyl group for G3 is Z/2 x Gl_3(\F_2), and there is a kind of exotic 2-complete symmetric space G3/Spin(7) whose mod two cohomology is concentrated in even degrees, with Euler characteristic 24 – see Th 1.8 of Dwyer-Wilkerson. The (mod 2) cohomology of G3 is an exterior algebra on generators of degree 7,11,13,14 so it’s (2-locally) a Poincar'e duality space of dimension 45. There’s also a recent relevant paper at
https://arxiv.org/abs/1903.10288
Nowadays I think people call such a thing a
https://en.wikipedia.org/wiki/P-compact_group
(eg for p=2); there’s a lot of systematic understanding of them, tho not by me. In particular I have a vague memory that G3 has a connection of some kind to a combinatorial construction studied by ?Louis Solomon and somebody but at the moment can’t find a reference…
BTW the Lie gp F_4, with its invariant framing, represents an interesting 52 = 2 x 26 - dimensional class in stable homotopy theory. I say this just to tease you.
best, hastily, Jack
Since I don’t feel that I have the leisure to do anything about this at the moment. I am forwarding this here in the hope that it inspires somebody to start a respective $n$Lab page!
Okay, started something based on this email, now at Dwyer-Wilkerson H-space.
Just a minimum, am being interrupted now…
(Also, should have merged the two threads now….)
Interesting! So this is the Dickson of Cayley-Dickson construction, presumably.
What’s the issue about 2-localness? Does this indicate the passage beyond rational homotopy theory to p-adic homotopy theory?
Interesting! So this is the Dickson of Cayley-Dickson construction, presumably.
More concretely: of the CY-construction on the octonions, hence of the sedenions, presumably.
I was just asking if the Dickson of Dickson invariant is the same person as that of Cayley-Dickson construction. And the answer is ’Yes’.
So then, as for Morava’s remark – “still waiting to be discovered by physicists”, what’s to be done? Is there some 2-adic brane bouquet which discovers $G_3$?
After all:
It was clear all along that the Cayley-Dickson construction knows something about supersymmetry and the stringy spacetimes, but this left open two problems: why consider star-algebras and their CD-doubles in the first place, and why stop the CD-process at some point?
Now with the bouquet, these two questions are answered. We see (that’s how I view it anyway) that those algebras are not the truly fundamental agent here. While they happen to neatly encode the crucial relations, the true fundamental concept is the progression of universal invariant (higher) central extensions of super Lie algebras. That this happens to be accompanied by division algebras for parts of the journey is a useful fact, but division algebras are not conceptually what drives this process.
Yeah, I don’t know.
So, what are the Dickson invariants, actually? Without knowing that, we can’t start to speculate about the relation of $G_3$ to physics.
I know I could discover the answer by looking through the references, but haven’t found the time yet…
Well we are told that the ring of mod 2 Dickson invariants of rank 4 is the ring of invariants of the natural action of $GL(4, \mathbf{F}_2)$ on the rank 4 polynomial algebra $H^{\ast}((B \mathbf{Z}/2)^4, \mathbf{F}_2)$, a polynomial algebra on classes $c_8$, $c_12$, $c_14$, and $c_15$ with $Sq^4 c_8 = c_{12}$, $Sq^2 c_{12} = c_{14}$, and $Sq^1 c_{14} = c_{15}$.
From the message by Jesper Grodal that just came in over AlgTop, p-compact Lie theory is pretty much ordinary Lie theory over Z/p. So maybe there wants to be some p-adic string theory, not sure.
Begin forwarded message by Jesper Grodal:
In fact, the classification of p-compact groups states that there is a 1-1-correspondence between isomorphism classes of connected p-compact groups, and isomorphism classes of root data over the p-adic integers (as conjectured by Clarence and others, in various forms, since the early days of the theory; I see my first email correspondence with Clarence on the subtleties of p=2 date from November 2000).
This is completely analogous to the classification of connected compact Lie groups, but replacing the integers Z by the p-adic integers Z_p.
Specializing to p=2 one gets as a corollary that any classifying space BX of a connected 2-compact group X splits as
BX \cong BG x BDI(4)^s
the product of the 2-completion of the classifying space of the compact Lie group G, and s copies of the Dwyer-Wilkerson space BDI(4) for some s.
DI(4) corresponds to the finite Z_2-reflection group which is number 24 on the Shepard-Todd list. It is the only irreducible finite complex reflection group which is realizable over Z_2 but not Z.
I wrote a survey of the classification and its history in my 2010 ICM talk, stressing the root data viewpoint: www.math.ku.dk/~jg/papers/icm.pdf
end of forwarded message
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