Also available at http://math.ucr.edu/home/baez/week193.html
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 (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 Dn 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 Dn
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 Dn" is interesting in any dimension, and it's
called Dn+. In 3 dimensions this is how carbon atoms are arranged in a
diamond! In any dimension, the volume of the unit cell of Dn+ is 1,
so we can say it's "unimodular". But Dn+ 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 x 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
for a total of
28 x 6 + 70 + 2 = 168 + 72 = 240
roots.
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:
e6
e4 e1
e7
e3 e2 e5
There should be 6 straight lines and a circle in your picture: 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 +-ei for i=0,...,7. Second, we include
(+- 1 +- ei +- ej +- ek)/2
whenever ei, ej and ek are imaginary octonions that all lie on the
same line in the above chart. Third, we include
(+- ei +- ej +- ek +- el)/2
whenever ei, ej, ek, and el are imaginary octonions that all lie *off*
the same line in the above chart.
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,
2^4 x 7 = 112 of the second form, and 2^4 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
are closed under multiplication - these are just the unit vectors among
the "Hurwitz integral quaternions", which form a D4 lattice in the
quaternions (see "week91"). So, each line in the Fano plane gives a
copy of the integral quaternions sitting inside the integral octonions.
Even better - I'm sorry, this is getting a bit technical, but I need to
write it down or I'll forget! - if we do the Cayley-Dickson construction
(see "week59") to any of these copies of the integral quaternions, we
get a bigger set of integral octonions that's also closed under addition
and multiplication. Unfortunately, this bunch is just a copy of D4 x D4
sitting inside E8, not the whole E8. E8 is the union of all these
D4 x D4's, one for line in the Fano plane. So, I have to calculate more
to finish convincing myself that the Cayley integral octonions are closed
under multiplication - or equivalently, that the 240 guys listed above
are closed under multiplication.
Anyway: this probably makes no sense to you, but *I'm* happy as a clam!
I can now see E8 as a bunch of octonions closed under addition and
multiplication, and I can calculate to my heart's content with them.
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).
If only the middle 3 of these subspace are nonzero, like so:
L = L(-1) + L(0) + L(1)
we say L is "3-graded". If only the middle 5 are nonzero, like so:
L = L(-2) + L(-1) + L(0) + L(1) + L(2)
we say L is "5-graded". And so on. In these situations, some
nice things happen.
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]
This product automatically satisfies two of identities defining
a Jordan algebra:
x o y = y o x
x o ((x o x) o y) = (x o x) o (x o y)
so 3-graded Lie algebras are a great source of Jordan algebras. Fourth
of all, in this situation L(0) acts on L(1) by means of the bracket
operation, so we get a Lie algebra of "infinitesimal symmetries" of our
Jordan algebra, too. Fifth of all, if L is 5-graded, we get a more
fancy algebraic structure called a "Kantor triple system", but I'm not
ready to talk about these, and you're probably not ready to listen,
either!
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.
Since I'm lazy, I figured out the number of roots with a "0"
as the first component by totalling up all the rest and subtracting
that from 240. That's how I got the number 84.
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)
as desired.
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
where I'm writing the dimension of each vector space direct below it.
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]
But we get a more exciting way of slicing the diamond if we use
the picture of E8 as the Cayley integral octonions! Let's do this,
and let each slice go through those roots where the "real part" x0
of our octonion
x0 + x1 e1 + ... + x7 e7
takes on some fixed value. This value can be 1, 1/2, 0, -1/2,
or -1, so we again get a 5-grading. Let's count the number of roots in
each slice:
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.
Here I got 56 roots with real part 1/2 by multiplying the number of lines
in the Fano plane by the number of sign choices in
(1 +- ei +- ej +- ek)/2
Similarly for the roots with real part -1/2. I got 126 roots with real
part 0 by subtracting all the other numbers on my list from 240.
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
since 126 + 8 = 134.
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
manifold on which E8 acts! In fact this manifold 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]
Here the "Freudenthal algebra" is the 56-dimensional irrep of E7,
which has an invariant symplectic structure and ternary product
satisfying some funky equations which get turned into the definition
of... a Freudenthal algebra!
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
Since E7's baby brother E6 is 78-dimensional, it's no surprise that
the Lie algebra L(0) is E6 plus u(1). Since 3-gradings tend to give
us Jordan algebras, it's no suprise that L(1) is the exceptional Jordan
algebra h_3(O) consisting of all 3x3 hermitian octonionic matrices.
E6 acts as the group of all transformations of h_3(O) preserving the
determinant, and in fact h_3(O) is an irrep of E6. L(-1) is just
the dual of this rep. So, we get a very octonionic description of E7:
E7 = h_3(O)* + [E6 + u(1)] + h_3(O)
Now, since E6 sits in E7 which sits in E8, just like nested Russian
dolls, we can take our previous description of E8:
E8 = [trivial] + [Freudenthal] + [E7 + u(1)] + [Freudenthal] + [trivial]
and decompose everything in sight as irreps of E6. If we do this, the
only new exciting thing that happens is that the Freudenthal algebra
decomposes into a copy of the exceptional Jordan algebra, a copy of its
dual, and two copies of the trivial rep:
[Freudenthal] = [trivial] + h_3(O)* + h_3(O) + [trivial]
At least I *think* this is right: people sometimes write elements
of the Freudenthal algebra as 2x2 matrices
a x
y b
where a,b are real and x,y lie in h_3(O), but I suspect they're
"cheating" a bit and identifying h_3(O) with its dual.
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]
Instead, we get transformations that tend to preserve a "determinant"
on L(1). People call L(0) the "structure algebra" of L(1) and call
the corresponding group the "structure group". There's a pretty
readable explanation here:
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.
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 R^{n-1} + R
so(n+2) so(n) n R^{n-1} + R
sp(2n) sl(n) (n^2+n)/2 h_n(R)
so(2n) sl(n) (n^2-n)/2 h_{n-1}(R)
sl(2n) sl(n)+sl(n) n^2 h_n(C)
so(4n) sl(2n) 2n^2-n h_n(H)
e_7 e_6 27 h_3(O)
e_6 so(10) 16 h_4(C)
Since L(0) always contains a u(1) summand in these cases, we
write
L(0) = L'(0) + u(1)
so that L'(0) is the interesting part of L(0). The formally
real simple Jordan algebras appearing here are all those listed
in "week162" - we get all of them! In particular, R^{n-1} + R is
the so-called "spin factor" Jordan algebra, which appears
in special relativity.
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,
Sel. 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,
/math.AG/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. Amer. Math. Soc. 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 Guenter 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.
-----------------------------------------------------------------------
Quote of the week:
The essential thing was that Serre each time strongly sensed the rich
meaning behind a statement that, on the page, would doubtless have
left me neither hot nor cold - and that he could "transmit" this
perception of a rich, tangible and mysterious substance - this
perception that is at the same time the desire to *understand*
this substance, to penetrate it.
Alexander Grothendieck, Recoltes et Semailles, p. 556.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html