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The Exceptional Lie Group E8 · 2007-03-19

a set of root vectors
The Lie algebra E8 is 248-dimensional: the 8-dimensional space depicted here, plus one dimension for each of the 240 root vectors.

(via bbc)


  1. — Melody    Mar 21, 14:39    #
    Please explain this math problem to me. What solution can it solved? Please give me some insight. Thanks!
  2. haken    Mar 21, 17:28    #
    Hm, first of all, I don’t know. I’ve chosen the picture for it is nice and easiely to be drawn by any arbitrary elementary school student … and when the pupil finished the task you can call it a set of root vectors. Thats basically why I posted it here. The matrix which was calculated is nothing that I ever tried to understand. Maybe that will change soon.

    However, up to now, I did not find an easy to understand explanation of the task – I am pretty sure that it wasn’t a classical math problem to calculate the E8-matrix, more a technical one for the sheer size of the thing – that was accomplished. ... and I am pretty sure that we won’t find this explanation, for there is no simple explanation. Thats why many scientists have spend a lot of time calculating. If one could get the whole idea from reading three paragraphs and looking at a picture there probably wouldn’t be so much fuzz about it.

    Now, for the implications … some scientists seem to hope, that now, when we know all the polynomes in the matrix, we can solve some fundamental problems in string theory (http://www.tenthdimension.com/medialinks.php) that have to do with symmetry ... but this is wishful thinking by now. Therefore I’m not sure whether the hype is justified.

    ... but … again, I do not know … I hardly understand what a Lie Group is … so, those who can give us more insight, please give us a hint. Comments are open.
  3. Haken    Mar 22, 18:48    #
    Here is a quite understandable piece on the topic: http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html

    and it comes with the following line, which is supported by a certain feeling deep in my guts:

    "Computing the Kazhdan–Lusztig–Vogan polynomials for E 8 is certainly nowhere nearly as important as the human genome project, nor as hard!"

  4. Haken    Mar 23, 16:26    #

    and this one is warmly recommended as well:

    http://www.math.columbia.edu/~woit/wordpress/?p=534

    While the calculation is a computational tour de force, and the computational methods may be useful elsewhere, the level of hype in the press releases, especially about the possible relations to physics, is somewhat disturbing.

    and for the basics this one too ...

    So, when you read this page, be SURE to realize that when I say "Lie group", that is my shorthand for "compact real form of a complex simple Lie group", and similar shorthand is being used when I say "Lie algebra".

    http://www.valdostamuseum.org/hamsmith/Lie.html

    I’m starting to gain some, the tiny little kina 'some', insight …

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