December 24, 2007

This Week's Finds in Mathematical Physics (Week 260)

John Baez

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This intro to Lie groups is also a bit quirky, but if you like Feynman diagrams or spin networks, it's irreplaceable:

25) Predrag Cvitanovic, Birdtracks, Lie's, and Exceptional Groups, available at http://www.nbi.dk/GroupTheory/

One of the great things about this book is that it classifies simple Lie groups according to their "skein relations" - properties of their representations, written out diagrammatically. In so doing, Cvitanovic realized that there's a "magic triangle" containing all the exceptional Lie groups. This subsumes the "magic square" of Freudenthal and Tits, which I discussed in "week145" and my octonion webpages.

This idea of Cvitanovic is closely related to the "exceptional series" of Lie groups - a pattern whose existence was conjectured by Deligne. I love the term "exceptional series". It's an oxymoron, since the exceptional groups were defined as those that don't fit into any series. But, it makes sense!

To see the exceptional series, it helps to do a mental backflip called "Tannaka-Krein duality", where you focus on the category of representations of the Lie group, instead of the group itself. Then, draw the morphisms in that category as diagrams, like Feynman diagrams! Then see what identities they satisfy. New patterns leap out: new series unify what had been "exceptions".

Very briefly, the idea goes like this. Suppose we have a Lie group G with Lie algebra L. The Lie bracket takes two elements x and y and spits out one element [x,y], and it's linear in each variable, so it gives a linear operator

L ⊗ L → L

which is actually a morphism in the category of representations of G.

So, following the philosophy of Feynman diagrams, we can draw the bracket operation like this:

                \     /
                 \   /
                  \ /
                   |
                   |
                   |
We can even use this to state the definition of a Lie algebra using diagrams! To say the bracket is antisymmetric:

[y,x] = -[x,y]

we just draw this:

                 \   /              |       |
                  \ /               |       |
                   /                |       |
                  / \               |       |
                 /   \               \     /
                 \   /     =    -     \   /
                  \ /                  \ /
                   |                    |
                   |                    |
                   |                    |
To say the Jacobi identity:

[x,[y,z]] = [[x,y],z] + [y,[x,z]]

we just draw this:

  \     \     /          \     /     /            \    /      /   
   \     \   /            \   /     /              \  /      /
    \     \ /              \ /     /                \       / 
     \     /                \     /                  \     /
      \   /        =         \   /         +       /  \   /
       \ /                    \ /                 /    \ /
        |                      |                  \     /
        |                      |                   \   /
        |                      |                    \ /
        |                      |                     |
If that's too cryptic, maybe this will explain what I'm doing:
 x     y       z        x       y     z          x      y      z
  \     \     /          \     /     /            \    /      /   
   \     \   /            \   /     /              \  /      /
    \     \ /              \ /     /                \       / 
     \     /                \     /                  \     /
      \   /        =         \   /         +       /  \   /
       \ /                    \ /                 /    \ /
        |                      |                  \     /
        |                      |                   \   /
        |                      |                    \ /
        |                      |                     |
   [x,[y,z]]              [[x,y],z]              [y,[x,z]]
But in fact, people usually massage this picture to make it even more cryptic, and call it the "IHX" identity - since the three terms look like the letters I, H, and X by the time they're done twisting them around. For a good explanation, with pretty pictures, see:

26) Greg Muller, Chord diagrams and Lie algebras, http://cornellmath.wordpress.com/2007/12/25/chord-diagrams-and-lie-algebras/

It then turns out that the exceptional Lie algebras F4, E6, E7 and E8 satisfy yet another identity:

 \        /                
  \      /                
   \----/                
   |    |                       
   |    |           = 
   /----\            
  /      \          
 /        \                
                       
 
  \        /           \   /
   \      /             \ /   
    \    /               |   
A    ----     +     A    |       +
    /    \               |  
   /      \             / \  
  /        \           /   \  


  \     /       \        /         \        /
   \   /         \      /           \      /
    \ /           \    /             \____/ 
B    /     +  B    |  |       +  B    ____    
    / \           /    \             /    \
   /   \         /      \           /      \
  /     \       /        \         /        \
for various choices of the constants A and B. So, they fit into a "series"!

I believe the main point of this identity, going back to Vogel's paper "Algebraic structures on modules of diagrams", is that for these Lie algebras, the square of the quadratic Casimir is the only degree-4 Casimir.

I think there's a lot more to be discovered here, in part by taking the gnarly computations people have done so far and making them more beautiful and conceptual. So, I urge all fans of exceptional mathematics, diagrams, and categories to look at these:

27) Pierre Deligne, La serie exceptionnelle des groupes de Lie, C. R. Acad. Sci. Paris Ser. I Math 322 (1996), 321-326.

Pierre Deligne and R. de Man, The exceptional series of Lie groups II, C. R. Acad. Sci. Paris Ser. I Math 323 (1996), 577-582.

Pierre Deligne and Benedict Gross, On the exceptional series, and its descendants, C. R. Acad. Sci. Paris Ser. I Math 335 (2002), 877-881. Also available as http://www.math.ias.edu/~phares/deligne/ExcepSeries.ps

28) Pierre Vogel, Algebraic structures on modules of diagrams, 1995. Available at http://www.institut.math.jussieu.fr/~vogel/ or http://citeseer.ist.psu.edu/469395.html

The universal Lie algebra, 1999. Available at http://www.institut.math.jussieu.fr/~vogel/

Vassiliev theory and the universal Lie algebra, 2000. Available at http://www.institut.math.jussieu.fr/~vogel/

For a good overview, try this:

28) J. M. Landsberg and L. Manivel, Representation theory and projective geometry, 2002. Available at arXiv:math/0203260.

Alas, they avoid drawing Feynman diagrams, though they talk about them in section 4. They prefer to use ideas from algebraic geometry:

29) J. M. Landsberg and L. Manivel, The projective geometry of Freudenthal's magic square, J. Algebra 239 (2001), 477-512. Also available as arXiv:math/9908039.

J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), 59-85. Also available as arXiv:math/0107032.

J. M. Landsberg and L. Manivel, Series of Lie groups, available as arXiv:math/0203241.

Bruce Westbury, whom longtime readers of This Week's Finds will remember as John Barrett's collaborator, has also worked on this subject. He has pointed out that both the magic square and the magic triangle can be given an extra row and column if we introduce a 6-dimensional algebra halfway between the quaternions and the octonions:

30) Bruce Westbury, Sextonions and the magic square, available as arXiv:math/0411428.

For even more references, try this:

31) Bruce Westbury, References on series of Lie groups, http://www.mpim-bonn.mpg.de/digitalAssets/2763_references.pdf

This stuff has been on my mind recently, since I've been working on exceptional groups and grand unified theories with my student John Huerta. Also, my friend Tevian Dray has a student who just finished a thesis on a related topic:

32) Aaron Wangberg, The structure of E6, available as arXiv:0711.3447.

In a nutshell: E6 is secretly SL(3,O). Octonions rock!


If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. - Thomas Jefferson


2007 John Baez
baez@math.removethis.ucr.andthis.edu