
For a while now I've been meaning to finish talking about monads and adjunctions, and explain what that has to do with the 4color theorem. But first I want to say a little bit more about "triality", which was the subject of "week90".
Triality is a cool symmetry of the infinitesimal rotations in 8dimensional space. It was only last night, however, that I figured out what triality has to do with 3 dimensions. Since it's all about the number three obviously triality should originate in the symmetries of threedimensional space, right? Well, maybe it's not so obvious, but it does. Here's how.
Take good old threedimensional Euclidean space with its usual basis of unit vectors i, j, and k. Look at the group of all permutations of {i,j,k}. This is a little 6element group which people usually call S_{3}, the "symmetric group on 3 letters".
Every permutation of {i,j,k} defines a linear transformation of threedimensional Euclidean space in an obvious way. For example the permutation p with
p(i) = j, p(j) = k, p(k) = idetermines a linear transformation, which we'll also call p, with
p(ai+ bj + ck) = aj + bk + ci.In general, the linear transformations we get this way either preserve the cross product, or switch its sign. If p is an even permutation we'll get
p(v) x p(w) = p(v x w)while if p is odd we'll get
p(v) x p(w) = p(v x w) = p(w x v).That's where triality comes from. But now let's see what it has to do with fourdimensional space. We can describe fourdimensional space using the quaternions. A typical quaternion is something like
a + bi + cj + dkwhere a,b,c,d are real numbers, and you multiply quaternions by using the usual rules together with the rules
i^{2} = j^{2} = k^{2} = 1, ij = k, jk = i, ki = j, ji = k, kj = i, ik = jNow, any permutation p of {i,j,k} also determines a linear transformation of the quaternions, which we'll also call p. For example, the permutation p I gave above has
p(a + bi + cj + dk) = a + bj + ck + di.The quaternion product is related to the vector cross product, and so one can check that for any quaternions q and q' we get
p(qq') = p(q)p(q')if p is even, and
p(q'q) = p(q')p(q)if p is odd. So we are getting triality to act as some sort of symmetries of the quaternions.
Now sitting inside the quaternions there is a nice lattice called the "Hurwitz integral quaternions". It consists of the quaternions a + bi + cj + dk for which either a,b,c,d are all integers, or all halfintegers. Here I'm using physics jargon, and referring to any number that's an integer plus 1/2 as a "halfinteger". A halfinteger is not any number that's half an integer!
You can think of this lattice as the 4dimensional version of all the black squares on a checkerboard. One neat thing is that if you multiply any two guys in this lattice you get another guy in this lattice, so we have a "subring" of the quaternions. Another neat thing is that if you apply any permutation of {i,j,k} to a guy in this lattice, you get another guy in this lattice  this is easy to see. So we are getting triality to act as some sort of symmetries of this lattice. And that is what people usually call triality.
Let me explain, but now let me use a lot of jargon. (Having shown it's all very simple, I now want to relate it to the complicated stuff people usually talk about. Skip this if you don't like jargon.) We saw how to get S_{3} to act as automorphisms and antiautomorphisms of R^{3} with its usual vector cross product... or alternatively, as automorphisms and antiautomorphisms of the Lie algebra so(3). From that we got an action as automorphisms and antiautomorphisms of the quaternions and the Hurwitz integral quaternions. But the Hurwitz integral quaternions are just a differently coordinatized version of the 4dimensional lattice D_{4}! So we have gotten triality to act as symmetries of the D_{4} lattice, and hence as automorphisms of the Lie algebra D_{4}, or in other words so(8), the Lie algebra of infinitesimal rotations in 8 dimensions. (For more on the D_{4} lattice see "week65", where I describe it using different, more traditional coordinates.)
Actually I didn't invent all this stuff, I sort of dug it out of the literature, in particular:
1) John H. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups, second edition, Grundlehren der mathematischen Wissenschaften 290, SpringerVerlag, 1993.
and
2) Frank D. (Tony) Smith, Sets and C^{n}; quivers and ADE; triality; generalized supersymmetry; and D4D5E6, preprint available as hepth/9306011.
But I've never quite seen anyone come right out and admit that triality arises from the permutations of the unit vectors i, j, and k in 3d Euclidean space.
I should add that Tony Smith has a bunch of farout stuff about quaternions, octonions, Clifford algebras, triality, the D_{4} lattice  you name it!  on his home page:
3) Tony Smith's home page, http://valdostamuseum.org/hamsmith/
He engages in more free association than is normally deemed proper in scientific literature  you may raise your eyebrows at sentences like "the Tarot shows the Lie algebra structure of the D4D5E6 model, while the I Ching shows its Clifford algebra structure"  but don't be fooled; his mathematics is solid. When it comes to the physics, I'm not sure I buy his theory of everything, but that's not unusual: I don't think I buy anyone's theory of everything!
Let me wrap up by passing on something he told me about triality and the exceptional groups. In "week90" I described how you could get the Lie groups G2, F4 and E8 from triality. I didn't know how E6 and E7 fit into the picture. He emailed me, saying:
"Here is a nice way: Start with D4 = Spin(8):28 = 28 + 0 + 0 + 0 + 0 + 0 + 0Add spinors and vector to get F4:52 = 28 + 8 + 8 + 8 + 0 + 0 + 0Now, "complexify" the 8+8+8 part of F4 to get E6:78 = 28 + 16 + 16 + 16 + 1 + 0 + 1Then, "quaternionify" the 8+8+8 part of F4 to get E7:133 = 28 + 32 + 32 + 32 + 3 + 3 + 3Finally, "octonionify" the 8+8+8 part of F4 to get E8:248 = 28 + 64 + 64 + 64 + 7 + 14 + 7This way shows you that the "second" Spin(8) in E8 breaks down as 28 = 7 + 14 + 7 which is globally like two 7spheres and a G2, one S7 for leftaction, one for rightaction, and a G2 automorphism group of octonions that is needed to for "compatibility" of the two S7s. The 3+3+3 of E7, the 1+0+1 of E6, and the 0+0+0 of F4 and D4 are the quaternionic, complex, and real analogues of the 7+14+7."
When I asked him where he got this, he said he cooked it up himself using the construction of E8 that I learned from Kostant together with the FreudenthalTits magic square. He gave some references for the latter:
4) Hans Freudenthal, Adv. Math. 1 (1964) 143.
5) Jacques Tits, Indag. Math. 28 (1966) 223237.
6) Kevin McCrimmon, Jordan Algebras and their applications, Bull. AMS 84 (1978) 612627, at pp. 620621. Available at http://projecteuclid.org
I would describe it here, but I'm running out of steam, and it's easy to learn about it from his web page:
7) Tony Smith, FreudenthalTits magic square, http://valdostamuseum.org/hamsmith/FTsquare.html
© 1996 John Baez
baez@math.removethis.ucr.andthis.edu
