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I am working on the next chapter in geometry of physics:
This is not done yet, but it should already be readable. To some extent I am taking the talk notes Super Lie n-algebra of Super p-branes (schreiber) and expand them into fully fledged lecture notes.
Since I am editing in a separate window, at this point I ask that everyone who feels like touching this page, even if it just concerns tiny changes (typos) to please alert me.
I am now working on polishing the next section “5. Branes”.
There is a fair bit of more material already in the entry, but it still needs more streamlining and harmonization.
With half an eye out for processes, like the brane bouquet, that iterate a procedure until they end in the exceptional, I see there’s something of this idea in sporadic simple groups:
the main induction step of the classification theorem was, that the centralizer of an involution of a simple group (an order-2-element/involution is the “only” thing we have “a-priori” in an arbitrary simple group by Feit-Thomson) is close (!) to simple there is the chance of sporadic group branching off from a Lie-Type or alternating group and inductively proceed for some steps until it terminates.
Pariahs are generated, along with ’roughly’
$M\leftarrow Co_1 \leftarrow M_{24}\leftarrow SL_3(4)$.
That the sporadics fall into two connected families (the Happy Family and the Pariahs) of subquotients is something incredibly interesting. Essentially one takes an extension of a smaller sporadic, then realises that extension of a (non-normal) subgroup of what turns out to be another sporadic. Though, from the cases I’ve ever looked at, sometimes one starts with the extension (arising naturally in some way, say as an automorphism group) and quotient down to the sporadic, then also embed the extension. I wonder if there is a ’maximally invariant cocycle’ things going on.
Thanks for the hint! This indeed sounds interesting. I should try to find time to look into this.
There’s a unique central extension of the binary Golay code which it is said gives rise to a construction of the monster.
Ah, but not an extension as a group, but as a symmetric 2-group. Conway only uses the Moufang loop structure on the objects of that 2-group, and really only to define a certain subgroup of the automorphisms of that loop. The structure of the Golay code as it relates to the Hamming code (see: $24 = 8\times 3$, and the construction here).
I really should write up the fact about that 2-group extension. I’d been hoping to get a student to work on it, but that would be at least a year away in my present case, if it all.
The most interesting thing would be the automorphism 2-group of this symmetric 2-group and how it relates to the construction of the monster…
Given the context of this discussion - extensions to higher super Lie algebras, an extension to a higher group would seem quite appropriate.
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