February 23, 2003

This Week's Finds in Mathematical Physics (Week 193)

John Baez

This is my last week in Sydney. The year-long drought in Australia has finally been broken by a series of rainstorms, but the sky was clear as I walked to my office tonight, and I saw the Milky Way really well! It's so much more prominent in the southern sky, since you can see the center of the Galaxy better.

Some issues of This Week's Finds are mainly for explaining things to other people, while others are mainly for myself. I'm afraid this Week is one of the latter. But I'll try to start by explaining what I'm up to.

Conversations with Tony Smith and Thomas Larsson have been making me think more about the biggest exceptional Lie group, that magnificent 248-dimensional monstrosity called E₈. This plays a significant role in string theory and some other attempts to wrap everything we know about physics into a big, glorious Theory of Everything. None of these attempts have succeeded in predicting anything new that's actually been observed (ahem), but I still think it's worth pondering the group E₈.

Why? First of all, it's a beautiful thing in itself. Second, it has strong ties to many "exotic" things in mathematics, including:

• the dodecahedron (see "week20" and "week65")

• the octonions (see "week141" and "week168")

• the Poincare homology 3-sphere (see "week163" and "week164")

• the 4-manifold called K3 and exotic smooth structures on R⁴ (see "week67")

• exotic spheres in 7 and 11 dimensions (see "week164")

... in short, a whole zoo of strange creatures! Third, if the laws of physics are indeed structures of "exceptional beauty" rather than "classical beauty" - see "week106" for an explanation of what I mean by that - then it's natural to hope that E₈ plays an important role.

How do we get our hands on E₈? It's a bit tricky. To understand a group, it's always best to see it as the symmetries of something. Often we try to see it as the symmetries of some vector space equipped with extra structure. But for E₈, the smallest vector space that will do the job is 248-dimensional - it's the Lie algebra of E₈ itself! In mathspeak, the smallest nontrivial irrep of E₈ is the adjoint rep.

But in normal Engish, the problem is this: it's hard to construct E₈ as the symmetries of anything simpler than itself. It reminds me of Baron von Munchausen pulling himself out of the swamp by his own bootstraps.

One possible way around this is to construct E₈ as the symmetries of something other than a vector space - for example, some manifold equipped with extra structure. Here there is some hope: the compact real form of E₈ is the isometry group of a 128-dimensional Riemannian manifold called the "octooctonionic projective plane". The reason for this name is that around 1956, Boris Rosenfeld claimed that you can construct this manifold as a projective plane over the "octooctonions": the octonions tensored with themselves. Unfortunately, while there's definitely something to this idea, I don't think anyone knows how to make it precise without first constructing E₈. Maybe someday....

Recently, some mathematical physicists have been studying a construction of E₈ as the symmetries of a 57-dimensional manifold equipped with extra structure:

1) Murat Gunaydin, Koepsell and Hermann Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys. 221 (2001), 57-76, also available as hep-th/0008063

2) Thomas A. Larsson, Structures preserved by exceptional Lie algebras, available as math-ph/0301006.

When I heard this, the number 57 instantly intrigued me - and not just because Heinz advertises "57 varieties", either! No, the reason is that the smallest nontrivial of irrep of E₈'s little brother E₇ is 56-dimensional: it's a vector space equipped with extra structure making it into the so-called "Freudenthal algebra". When you study this subject long enough, you realize that strange numbers can serve as clues to hidden relationships... and guess what: there's one here! I'll say a bit more about it later.

(By the way, the story behind Heinz's "57 varieties" is that Henry John Heinz saw an ad for 21 styles of shoe, and liked the gimmick - but the numbers 5 and 7 held a special significance for him and his wife. If you don't believe me, send a letter to Heinz Consumer Affairs, P.O. Box 57, Pittsburgh, PA 15230 and ask them!)

Another way to get ahold of the group E₈ is starting with its "root lattice", the so-called E₈ lattice. There are different ways to describe this. Perhaps the most efficient is to say that it's the densest lattice packing of spheres in 8 dimensions! If I were about to drown and needed to define the E₈ lattice before I went under, this is how I'd do it. Unfortunately this leaves the recipient of the message with a lot of work: they have to find the lattice meeting this description.

A more user-friendly description is this. In any dimension we can make a "checkerboard" with alternating red and black hypercubes, and we get a lattice by taking the centers of all the red ones. In n dimensions this is called the D_n lattice. We can pack spheres by centering one at each point of this lattice and making them just big enough so they touch. There will of course be some space left over. But when we get up to dimension 8, there's enough room left over so we can slip another identical array of spheres in the gaps between the ones we've got! This gives the E₈ lattice.

We can translate this into formulas without too much work. The D_n lattice consists of all n-tuples of integers that sum to an even integer: requiring that they sum to an even integer picks out the center of every other hypercube in our checkerboard. Then, to get E₈, we take the union of two copies of the D₈ lattice: the original one and another one shifted by (1/2, ..., 1/2).

(Actually this "doubled D_n" is interesting in any dimension, and it's called D_n⁺. In 3 dimensions this is how carbon atoms are arranged in a diamond! In any dimension, the volume of the unit cell of D_n⁺ is 1, so we can say it's "unimodular". But D_n⁺ is only a lattice in even dimensions. In dimensions that are multiples of 4, it's an "integral" lattice, meaning that the dot product of any two vectors in the lattice is an integer. And in dimensions that are multiples of 8, it's also "even", meaning that the dot product of any vector with itself is even. In fact, even unimodular lattices are only possible in Euclidean space when the dimension is a multiple of 8. D₈⁺ = E₈ is the only even unimodular lattice in 8 dimensions; in 16 dimensions there are just two: E₈ × E₈ and D₁₆⁺. As explained in "week95", these give two versions of heterotic string theory.)

Summarizing, we can say E₈ consists of all 8-tuples of real numbers (x₁, ..., x₈) that sum to an even integer and that are either all integers or all integers plus 1/2.

Using this description it's easy to see that when you pack spheres in an E₈ lattice, each sphere touches 240 others. The reason is that the shortest nonzero vectors in this lattice, the so-called "roots", have length-squared equal to 2, and there are 240 of them:

(1,1,0,0,0,0,0,0) and all permutations thereof:
there are 8 choose 2 = 28 of these

(-1,-1,0,0,0,0,0,0) and all permutations thereof:
there are 8 choose 2 = 28 of these

(1,-1,0,0,0,0,0,0) and all permutations thereof:
there are twice 8 choose 2 = 56 of these

(1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2):
there is 1 of these 

(-1/2,-1/2, 1/2,1/2,1/2,1/2,1/2,1/2):
there are 8 choose 2 = 28 of these

(-1/2,-1/2,-1/2,-1/2, 1/2,1/2,1/2,1/2):
there are 8 choose 4 = 70 of these

(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2, 1/2,1/2): 
there are 8 choose 2 = 28 of these

(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2): 
there is 1 of these

28 x 6 + 70 + 2 = 168 + 72 = 240

There's also another description of the E₈ lattice, which I've been meaning to understand for ages, but which always scared me. You can think of 8-dimensional space as the octonions. The unit octonions are closed under multiplication and taking inverses. If you take the E₈ lattice, rescale it so the roots have length one, and rotate it correctly, you get a collection of 240 unit octonions that are closed under multiplication! It then follows that the octonions in the E₈ lattice are closed under addition and multiplication; these are called the "Cayley integral octonions".

This sounds like just the sort of thing I'd like; the problem is the phrase "rotate it correctly". First, you have to rotate the rescaled E₈ lattice so that it contains the octonion 1. That already means that the coordinate system used above is not the one we usually use for octonions, where

(x₀,...,x₇) = x₀ + x₁ e₁ + ... + x₇ e₇

with e₁,...,e₇ being the unit imaginary octonions, which we multiply using the standard octonion multiplication table. And just rotating the lattice any old way so that it contains 1 = (1,0,0,0,0,0,0,0) is not good enough; you have to do it the right way to get a lattice closed under multiplication.

The right way is described in Conway and Sloane's book (see "week20"). These days you can even look it up on the web:

3) Neil J. A. Sloane, Index of Lattices, the E₈ lattice: coding version, http://www.research.att.com/~njas/lattices/E₈_code.html

However, it always scared me, because the description involved the "Hamming code H(8,4,4)". You see, lattices are closely connected to coding theory - not coding in the sense of cryptography, but coding in the sense of efficient data transmission. In a code like this you want to pack information as efficiently as possible while keeping some error-correction ability, and mathematically this is related to the problem of densely packing spheres in higher-dimensional space! This is all very cool, but I don't understand it very well... and more importantly, whenever I looked at the description of the Hamming code H(8,4,4), I could "understand it" in the sense of nodding in mute assent, but not in the sense of seeing how it was related to anything.

Luckily, I now see how to get around this. Instead of describing the Cayley integral octonions using the theory of codes, I now see how to describe them using the octonion multiplication table! I'm sure everyone else already knew this - but they never told me.

Here's how it goes. First you have to remember your multiplication table - the octonion multiplication table, that is. Draw an equilateral triangle, draw a line from each corner to the midpoint of the opposite side, and inscribe a circle in the triangle. Then label the corners, the midpoints of the edges and the center of the triangle with the unit imaginary octonions, any way you like:

There are 6 straight lines and a circle here: we call these all "lines", and call this gadget the "Fano plane". There are 7 points and 7 lines: each point lies on 3 lines, and each line goes through 3 points... very nice.

I won't describe how to use this picture to multiply octonions, since I already did that in "week104", and we won't need that here.

Now let me describe the Cayley integral octonions. I'll actually describe all 240 of them that have length 1. Integer linear combinations of these give the Cayley integral octonions - or in other words, a rescaled version of the E₈ lattice.

First, we include ±e_i for i=0,...,7. Second, we include

(± 1 ± ei ± ej ± ek)/2

(± ei ± ej ± ek ± el)/2

It's easy to see that all these octonions have length 1. It's also easy to count them! There are 2 x 8 = 16 of the first form, 2⁴ x 7 = 112 of the second form, and 2⁴ x 7 = 112 of the third form, for a total of 240.

It's harder to check that these 240 guys are closed under multiplication. You can save some work by noticing that each line in the Fano plane gives a copy of the quaternions sitting inside the octonions. Moreover, the 24 quaternions of the form

±1, ±i, ±j, ±k,   (± 1 ± i ± j ± k)/2

[Note: I later realized that they are not closed under multiplication! We have a perfectly fine E₈ lattice, so everything that follows is okay... but it's not the Cayley integral octonions! I'll explain this next week.]

Anyway: this probably makes no sense to you, but I'm happy as a clam! So what can I do with them, for example?

Well, I can see some ways to make E₈ into a graded Lie algebra!

I guess I should start by saying some general stuff about graded Lie algebras, which explains why this is interesting.

For starters, I'm not talking about Z/2-graded Lie algebras, also known as "Lie superalgebras"; I'm talking about taking a plain old Lie algebra L and writing it as a direct sum of subspaces L(i), one for each integer i, such that

[L(i), L(j)] is contained in L(i+j).

L = L(-1) ⊕ L(0) ⊕ L(1)

L = L(-2) ⊕ L(-1) ⊕ L(0) ⊕ L(1) ⊕ L(2)

First of all, L(0) is always a Lie subalgebra of L. Second of all, it acts on each other space L(i) by means of the bracket. Third of all, if L is 3-graded, we can give L(1) a product by picking any element k of L(-1) and defining

x o y = [[x,k],y]

x o y = y o x

x o ((x o x) o y) = (x o x) o (x o y)

There's a lot more to say about this stuff, but let's just see a bit about how it works for E₈. We've got two nice pictures of the 240 roots of the E₈ lattice; you should imagine these as the dazzling vertices of a beautiful diamond in 8 dimensions. To get a grading on E₈, all we need to do is slice this diamond with evenly spaced parallel hyperplanes in such a way that each vertex of the diamond, as well as its center, lies on one of these hyperplanes. There are different ways to do this, so you should imagine yourself as a gem cutter, turning around this diamond, looking for nice ways to slice it.

For example, if we use our picture of the E₈ lattice as 8-tuples that sum to an even integer are either all integers or all half-integers, one obvious way to slice the diamond is to let each slice go through those roots where the first coordinate takes on some fixed value. The first coordinate can be 1, 1/2, 0, -1/2, or -1, so we get a 5-grading. Let's work out how many roots there are of each kind:

The number of roots with a "1" as the first component is
7 + 7 = 14.

The number of roots with a "1/2" as the first component is
1 + (7 choose 5) + (7 choose 3) + (7 choose 1) = 1 + 21 + 35 + 7 = 64.

The number of roots with a "0" as the first component is 84.

The number of roots with a "-1/2" as the first component is
1 + (7 choose 5) + (7 choose 3) + (7 choose 1) = 1 + 21 + 35 + 7 = 64.

The number of roots with a "-1" as the first component is
7 + 7 = 14.

Now, whenever you have a simple Lie algebra it's a direct sum of "root spaces", one for each root, together with an n-dimensional subspace called the Cartan algebra, where n is the called the "rank" of the Lie algebra. The rank of E₈ is 8, so its dimension is 240 + 8 = 248. When we taking our way of slicing the diamond and convert it into a grading of E₈, the roots in the ith slice form a basis of L(i), except we also have to count the Cartan as part of L(0). Thus in this example the dimension of L(0) is not just 84 but 84 + 8 = 92. Some basic stuff about simple Lie algebra guarantees that this trick always works: we get

[L(i), L(j)] is contained in L(i+j)

So, in this example we get a 5-grading where

E8 =  L(-2) ⊕ L(-1) ⊕ L(0) ⊕ L(1) ⊕ L(2)
248 =  114  +  64   +  92   +  64  +  14

Now, L(0) is a Lie algebra, but which one? To figure this out we need to think about how this diamond-cutting trick worked. At least in this case - and in fact it often works like this - the roots in the 0th slice are just the roots of a simple Lie algebra of rank one less than the one we started with. Since the Cartan of this smaller Lie algebra is one dimension smaller, it turns out that L(0) equals this smaller Lie algebra plus a one-dimensional abelian subalgebra - namely u(1).

In this example this smaller Lie algebra is so(14), which has dimension 91. L(1) is a 64-dimensional chiral spinor rep of so(14), and L(2) is the 14-dimensional vector rep... and similarly for L(-1) and L(-2). So we get a very "14-dimensional" picture of E₈:

E8 =  [vectors] ⊕ [spinors] ⊕ [so(14) ⊕ u(1)] ⊕ [spinors] ⊕ [vectors]

x0 + x1 e1 + ... + x7 e7

The number of roots with real part 1 is 1.

The number of roots with real part 1/2 is 56.

The number of roots with real part 0 is 126.

The number of roots with real part -1/2 is 56.

The number of roots with real part -1 is 1.

(1 ± ei ± ej ± ek)/2

So, we get a 5-grading of E₈ like this:

E8 =  L(-2) ⊕ L(-1) ⊕ L(0) ⊕ L(1) ⊕ L(2)
248 =   1   +  56   +  134  +  56  +  1

This shows how to get E₈ to act on a 57-dimensional manifold: we form the group E₈, and form the subgroup G whose Lie algebra is L(-2) ⊕ L(-1) ⊕ L(0), and the quotient E₈/G will be a 57-dimensional space on which E₈ acts! In fact this space is the smallest "Grassmannian" of E₈, as explained in "week181" - look at the picture of the E₈ Dynkin diagram near the end.

My goal in life is now to define a set of algebraic varieties, one for each root in L(1) and L(2), so I can write a paper entitled "57 Varieties" and get sued for trademark infringement by Heinz.

In the above grading of E₈, the Lie algebra L(0) is the direct sum of E₇ and u(1). This is no surprise if you know that the dimension of E₇ is 133... but the reason it's true is that if you take the roots of E₈ that are orthogonal to any one root, you get the roots of E₇. So, we get a very E₇-ish description of E₈:

E8 =  [trivial] ⊕ [Freudenthal] ⊕ [E7 ⊕ u(1)] ⊕ [Freudenthal] ⊕ [trivial]

There are a lot of other games we can play like this, but like solitaire they're not too fun to watch, so I'll just mention one more, and then give a bunch more references.

Above we have seen the roots of E₇ as the imaginary Cayley integral octonions of norm 1. These form a 7-dimensional gemstone with 126 vertices, and we can repeat the same "gem-slicing" trick on a smaller scale to get gradings of the Lie algebra E₇. If we do this in a nice way, we get a 3-grading of E₇:

E7  =  L(-1) ⊕ L(0) ⊕ L(1)
133 =    27  +  79  +  27

E7 = h3(O)* ⊕ [E6 ⊕ u(1)] ⊕ h3(O)

E8  =  [trivial] ⊕ [Freudenthal] ⊕ [E7 + u(1)] &oplus [Freudenthal] ⊕ [trivial]

[Freudenthal] = [trivial] ⊕ h3(O)* ⊕ h3(O) ⊕ [trivial] 

a x 
y b

In short, E₈ contains a lot of other "exceptional" structures, all arranged in a very nice way.

Now for some references and apologies.

I didn't do justice to the stuff about Jordan algebras and 3-graded Lie algebras, because I'm still confused about certain aspects. For example, where does the unit in the Jordan algebra come from? I also didn't explain precisely what sort of "infinitesimal symmetries" we get from the action of L(0) on L(1). If we exponentiate these infinitesimal symmetries, we don't usually get automorphisms of L(1), since there's no reason for the element "k" to preserved - remember that

x o y = [[x,k],y]

4) Kevin McCrimmon, Jordan Algebras and their applications, Bull. AMS 84 (1978) 612-627.

and hopefully even more here:

5) Kevin McCrimmon, A Taste of Jordan Algebras, Springer, Berlin, perhaps to appear in March 2003. Available for free online at http://math1.uibk.ac.at/mathematik/jordan/archive/atoja/ - but watch out, it's 545 pages long!

In fact, all this is part of a bigger relationship between 3-graded Lie algebras and so-called "Jordan triple systems" known as the Tits-Kantor-Koecher construction. Jordan triple systems are a generalization of Jordan algebras - and I'm sort of confused about why this generalization also turns up here. I guess I should read these too:

6) J. Tits, Une class d'algebres de Lie en relations avec les algebres de Jordan, Ned. Akad. Wet., Proc. Ser. A 65 (1962), 530.

7) M. Koecher, Imbedding of Jordan algebras into Lie algebra I, Am. J. Math. 89 (1967), 787.

8) Soji Kaneyuki, Graded Lie algebras, related geometric structures, and pseudo-hermitian symmetric spaces, in Analysis and Geometry on Complex Homogeneous Domains, by Faraut, Kaneyuki, Koranyi, Lu, and Roos, Birkhauser, New York, 2000.

Kaneyuki has made some nice tables of 3-gradings on simple Lie algebras, and you can see some of these here:

9) Tony Smith, Graded Lie algebras, http://www.innerx.net/personal/tsmith/GLA.html

Thomas Larsson has made a nice table of all the formally real simple Jordan algebras you get from 3-graded simple Lie algebras, and here it is, slightly modified:

Lie algebra L   L'(0)           dim(L(1))      Jordan algebra L(1)

sl(n+1)         sl(n)           n               Rn-1 ⊕ R
so(n+2)         so(n)           n               Rn-1 ⊕ R
sp(2n)          sl(n)           (n2+n)/2        hn(R)
so(2n)          sl(n)           (n2-n)/2        hn-1(R)
sl(2n)          sl(n)+sl(n)     n2              hn(C)
so(4n)          sl(2n)          2n2-n           hn(H)
E7               E6             27              h3(O)
E6               so(10)         16              h4(C)

L(0) = L'(0) + u(1)

For the more intricate relationship between 5-graded Lie algebras, Freudenthal algebras and Kantor triple systems, I should reread these:

10) I. Kantor, I. Skopets, Some results on Freudenthal triple systems, 10el. Math. Sov. 2 (1982), 293.

11) K. Meyberg, Eine Theorie Der Freudenthalschen Tripelsysteme, I, II, Ned. Akad. Wet., Proc. Ser. A 71 (1968), 162-190.

12) R. Skip Garibaldi, Structurable algebras and groups of types E₆ and E₇, available at math.RA/9811035.

13) R. Skip Garibaldi, Groups of type E₇ over arbitrary fields, available at math.RA/9811056.

14) G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Class. Quant. Grav. 4 (1987), 227-236.

Also, I didn't say anything yet about the connection of Lie triple systems, Jordan algebras, and Jordan triple systems to the geometry of symmetric spaces! There is in fact a dictionary relating these funny algebraic structures to very nice kinds of geometry, which motivates the Tits-Kantor-Koecher construction and its generalizations. Someday I may understand this well enough to explain it. For now, you should try to get ahold of these:

15) W. Bertram, The Geometry of Jordan and Lie structures, Lecture Notes in Mathematics 1754, Springer, Berlin, 2001.

16) Ottmar Loos, Jordan triple systems, R-spaces and bounded symmetric domains, Bull. AMS 77 (1971), 558-561.

17) Ottmar Loos, Symmetric Spaces I: General Theory, W. A. Benjamin, New York, 1969. Symmetric Spaces II: Compact Spaces and Classification, W. A. Benjamin, New York, 1969.

Unfortunately of the last two books I can get only volume I at U.C. Riverside, and only volume II here at Macquarie University! Someone should reprint both of these books: they're nice. Loos has also written a book on "Jordan pairs", but in my current state of development I find that unreadable.

Addendum: Blichfeldt proved in 1935 that E₈ is a maximally dense lattice packing of spheres in 8 dimensions, and Vetcinkin proved in 1980 that it's the unique lattice packing that achieves this density in 8 dimensions. Now Cohn and Kumar have shown that the E₈ packing is darn close to the densest of all sphere packings in 8 dimensions, lattice or not. No other can be more than 1 + 10^-14 as dense as this one!

They also showed that in 24 dimensions no packing can be more than 1 + 10^-29 times as dense as the Leech lattice, and that this is the unique best lattice packing. Of course the E₈ and Leech lattices are probably the best of all sphere packings in their dimensions, but it's very hard to understand the set of all sphere packings, so even these partial results are amazing.

Here are their papers:

18) H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, available at math.MG/0403263.

H. Cohn and A. Kumar, The densest lattice in twenty-four dimensions, Elec. Res. Ann. 10 (2004), 58-67. Available online at http://www.ams.org/era/2004-10-07/S1079-6762-04-00130-1/

There's also a really nice overview of this topic in the American Mathematical Society Notices, which explains how people manage to prove results about all packings:

19) Florian Pfender and Günter M. Ziegler, Kissing numbers, sphere packings, and some unexpected proofs, AMS Notices 51 (September 2004), 873-883. Available online at http://www.ams.org/notices/200408/200408-toc.html

And while you're at it, read this article, which studies a question mentioned in "week20":

20) Bill Casselman, The difficulties of kissing in three dimensions, AMS Notices 51 (September 2004), 884-885. Available online at http://www.ams.org/notices/200408/200408-toc.html

namely, how to roll twelve balls in 3 dimensions around the surface of a thirteenth ball of equal size.

© 2003 John Baez
baez@math.removethis.ucr.andthis.edu