Questions
Wednesday 2007 May 30
This is an experimental page where people can leave questions, that I may or may not answer.
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Thursday 2007 May 31 at 5:29 pm
Thank you for providing this space for questions. I would like to ask the following one: in What Is The Monster you write that “probably the best answer” is that it’s the automorphism group of the monster vertex algebra, yet adding that none of the four possible definitions are completely satisfactory.
To a beginner this sounds intringuing, since simple groups are very natural objects to define, whereas vertex algebras seem more convoluted. So the catch must be in the notion of being sporadic: did you imply that while some simple groups will appear to us as sporadic it will mean we miss something (and if so is that for aesthetic reasons or are there mathematical clues), or am I misunderstanding the issue?
Friday 2007 June 1 at 2:05 am
The problem is that although simple groups are natural objects, they can be very hard to construct. In particular for the monster group (and for many of the other sporadic groups) there is no easy construction known. And for some of the sporadic groups there is no good explanation of why they exist: they just appear at the conclusion of a very long, complicated, and mysterious calculation.
Friday 2007 June 1 at 1:57 pm
The “Expermental Mathematics” thread of this great blog asks whether or not the 26 sporadic groups are a “kabbalistic coincidence” or the hint of some deeper Math. Richard Borcherds is right that the sporadics, historically, appeared mysteriously. The history of mathematics has many examples of objects that were mysterious when first discovered, and later shown to be “natural” or “inevitable” or “inescapable.”
Assuming that human beings do not destroy themselves, the millennia of Math History will be a smaller and smaller part of what is known. If the future “singularity” of science fiction is valid, mathematical understanding will increase exponentially.
Tuesday 2007 June 26 at 7:00 pm
Actually it’s more a request than a question
My name is Saad and I’m working on a gravity model
I think that mass explanation is something simple and reachable. but I’m facing only one problem! The Newton force law is wrong with small distances and high masses. And there is no other verified referential!
I can not build a model based on a wrong gravity approximation
In my website I propose a new gravitational approximation and I would like to have your opinion on it.
I hope you will have time to visit it
thank you
Monday 2007 July 9 at 2:18 am
Dear Richard Borcherds,
I have found delightful structures hidden in the count of the number of letters in the names of numbers in given languages. My question is, though, did you and Neil J. A. Sloane author A051785 Number of letters in n-th Catalan number. as a kind of pun on the classical term “Catalan numbers” in the sense of Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.?
Friday 2007 July 27 at 3:59 am
The bad joke about Catalan numbers is due to Richard Stanley, and occurs as one of the exercises in his textbooks on combinatorics.
Wednesday 2007 August 15 at 10:21 am
Does anyone know if there is a nice model for the classifying spaces of the exceptional simple Lie groups? i.e. something like the grassmannian models for the classical Lie groups. I couldn’t find it in any of the books and I can’t get the question out of my head.
(Stronger question: Is there a *uniform* construction, starting with the Dynkin diagrams, of classifying spaces for all the simple Lie groups? I haven’t even managed to unify the grassmannian constructions for the classical groups from the point of view of Dynkin diagrams.)
Or is this one of those well-known questions everyone knows not to waste their time on?
Tuesday 2007 December 25 at 6:46 am
Sorry for the very delayed reply. I don’t know offhand of any nice model for classifying spaces of exceptional groups. There is a uniform construction of classifying spaces for the Dynkin diagrams as there are several known functorial ways to construct a classifying space from the group, but none of them are all that nice: you construct some really large constructible space acted on freely by the group and take the quotient.
Thursday 2008 February 21 at 12:34 pm
Dear Richard Borcherds,
I am a graduate student in physics. I have a few doubts on Lorentzian Kac-Moody algebras in general.
1) The simple roots of affine Lie algebras have the simple roots of the horizontal sub algebra, as well as an additional root of the form
[ \alpha_0 = \delta - (highest root of horizontal sub algebra) ].
where delta is the smallest imaginary root.
Do the Lorentzian GKMs have a root which is of the above form?
The imaginary roots (a,a) = 0, are generated as multiples of the root (\delta). It is not very clear to me how the roots of the form (a,a)< 0 are generated.
2) The set of positive roots of affine Lie algebras contains all the roots of the horizontal algebra, and roots of the form ( \alpha_i + n. \delta ), where the \alpha_i belong to the root space of the horizontal sub algebra, and \delta are roots of zero norm.
Are there roots of this form in the Lorentzian case? Are there roots of the form (\alpha_i + n. \beta), for \beta being a root of negative norm?
3) The Weyl group of an affine Lie algebra is the semi-direct product of the Weyl group of its horizontal sub algebra and of a freely generated abelian group of translations (on the co-root lattice). Does such a realization exist in the Lorentzian case?
Friday 2008 February 22 at 5:50 am
1. No, GKMs do not usually have anything like a highest root.
2. Probably yes, though the question is a bit unclear.
3. No, the Weyl group of a GKM in general has no decomposition as a semidirect product. The interesting ones tend to have a Weyl group that is a hyperbolic reflection group.
Monday 2008 February 25 at 8:27 am
Thanks for the answers. You forgot the bit about the (a,a)< 0 roots. How are the generated?
For example in the case of the monster Lie algebra, all the imaginary roots are of the form (n. \rho.) (with multiplicity 24), where rho is the norm zero root. Is there something similar for the (a,a)<0 roots? How does one determine them exactly?
In the denominator identity the exponential in the summation side consists of two terms the second of which is a summation over the simple roots whose norm is less than zero. Is there a way to compute the terms explicitly, in principle atleast, like Lepowski and Milne do for the case of affine Lie algebras ?
Thanks once again for the previous answers.
Saturday 2008 July 5 at 12:46 pm
thanks for providing such a nice explanation