The Unapologetic Mathematician

Mathematics for the interested outsider

A rough overview

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March 22, 2007 - Posted by | Atlas of Lie Groups


  1. I am not a mathematician but still I try to comprehend. Are there 2 or 3 dimensional diagrams of these simple Lie groups?

    Comment by eliza | March 24, 2007 | Reply

  2. Well, sort of. The lower ones down, at least. For example, B1 is the collection of all rotations of three-dimensional space. In general, Bn is made up of rotations in (2n+1)-dimensional space, and the D series gives rotations in even-dimensional spaces. Unfortunately, due to a technicality, rotations in the plane aren’t considered a simple Lie group.

    If you’re thinking of a diagram like the one that ran alongside all the news reports, it’s actually not the Lie group they’re talking about. It’s a sort of tool used in Cartan’s classification called a “root system”, and the picture is a 2-dimensional rendering of an 8-dimensional (for E8) shape. There are pictures like these for all Lie groups, and John Baez has a bunch of them in his most recent column.

    Comment by John Armstrong | March 24, 2007 | Reply

  3. Can you please dumb it down further and explain any possible practical application for this? Maybe cite something Star Trek or Star Wars and a maybe a reference to a weapon or some cool space ship? Or even cooler some invading alien force? I understand what your saying but I have no real frame of reference for it because I am not a mathematician but I do understand stuff like “fundamental underlying principle to teleporation” or “fundamental underlying principle to big explosions”. Or even “fundamental underlying principle for making space craft fly”.

    I just don’t have a frame of reference for this formula or how it matters.

    Comment by Kenny Coffin | March 25, 2007 | Reply

  4. Kenny, that’s a great question and it deserves its own post. I’m going to mull it over and write it up in the next day or so.

    Comment by John Armstrong | March 25, 2007 | Reply

  5. Cool! Thanks for the explanation. Seems like the press could certainly have explained that. So can this be applied to computer graphics?

    Comment by SomeGuy | March 26, 2007 | Reply

  6. I’m actually not sure what this can directly apply to. I just like it ’cause it’s pretty (in an intellectual sense). I’ve just put up another post linking to other sketches by the people directly involved, and my thoughts on why (in a real-world sense) we care about this.

    Comment by John Armstrong | March 26, 2007 | Reply

  7. […] I gave a quick overview of the idea of a Lie group, a Lie algebra, and a representation; a rough overview for complete neophytes of what, exactly, had been calculated; and an attempt to explain why we […]

    Pingback by Lie Groups in Nature « The Unapologetic Mathematician | January 11, 2010 | Reply

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