
Well, it seems I want to talk one more time about octonions before moving on to other stuff. I'm a bit afraid this obsession with octonions will mislead the nonexperts, fooling them into thinking octonions are more central to mainstream mathematical physics than they actually are. I'm also worried that the experts will think I'm spend all my time brooding about octonions when I should be working on practical stuff like quantum gravity. But darn it, this is summer vacation! The only way I'm going to keep on cranking out "This Week's Finds" is if I write about whatever I feel like, no matter how frivolous. So here goes.
First of all, let's make sure everyone here knows what projective space is. If you don't, I'd better explain it. This is honest mainstream stuff that everyone should know, good nutritious mathematics, so I won't need to feel too guilty about serving the extravagant octonionic dessert which follows.
Start with R^{n}, good old ndimensional Euclidean space. We can imagine wanting to "compactify" this so that if you go sailing off to infinity in some direction you'll come sailing back from the other side like Magellan. There are different ways to do this. A wellknown one is to take R^{n} and add on one extra "point at infinity", obtaining the ndimensional sphere S^{n}. Here the idea is that start anywhere in R^{n} and start sailing in any direction, you are sailing towards this "point at infinity".
But there is a sneakier way to compactify R^{n}, which gives us not the ndimensional sphere but "projective nspace". Here we add on a lot of points, one for each line through the origin. Now there are lots of points at infinity, one for every direction! The idea here is that if you start at the origin and start sailing along any straight line, you are sailing towards the point at infinity corresponding to that line. Sailing along any parallel line takes you twoards the same point at infinity. It's a bit like a perspective picture where different families of parallel lines converge to different points on the horizon  the points on the horizon being points at infinity.
Projective nspace is also called RP^{n}. The R is for "real", since this is actually "real projective nspace". Later we'll see what happens if we replace the real numbers by the complex numbers, quaternions, or octonions.
There are some other ways to think about RP^{n} that are useful either for visualizing it or doing calculations. First a nice way to visualize it. First take R^{n} and squash it down so it's just the ball of radius 1, or more precisely, the "open ball" consisting of all vectors of length less than 1. We can do this using a coordinate transformation like:
x → x' = x/(1 + x^{2})^{½}
Here x stands for a vector in R^{n} and x is its length. Dividing the vector x by (1 + x^{2})^{½} gives us a vector x' whose length never quite gets to 1, though it can get as close at it likes. So we have squashed R^{n} down to the open ball of radius 1.
Now say you start at the origin in this squashed version of R^{n} and sail off in any direction in a straight line. Then you are secretly heading towards the boundary of the open ball. So the points an the boundary of the open ball are like "points at infinity".
We can now compactify R^{n} by including these points at infinity. In other words, we can work not with the open ball but with the "closed ball" consisting of all vectors x' whose length is less than or equal to 1.
However, to get projective nspace we also have to decree that antipodal points x' and x' with x' = 1 are to be regarded as the same. In other words, we need to "identify each point on the boundary of the closed ball with its antipodal point". The reason is that we said that when you sail off to infinity along a particular straight line, you are approaching a particular point in projective nspace. Implicit in this is that it doesn't matter which way you sail along that straight line. Either direction takes you towards the same point in projective nspace!
This may seem weird: in this world, when the cowboy says "he went thataway" and points at a particular point on the horizon, you gotta remember that his finger points both ways, and the villian could equally well have gone in the opposite direction. The reason this is good is that it makes projective space into a kind of geometer's paradise: any two lines in projective space intersect in a single point. No more annoying exceptions: even "parallel" lines intersect in a single point, which just happens to be a point at infinity. This simplifies life enormously.
Okay, so RP^{n} is the space formed by taking a closed ndimensional ball and identifying pairs of antipodal points on its boundary.
A more abstract way to think of RP^{n}, which is incredibly useful in computations, is as the set of all lines through the origin in R^{n+1}. Why is this the same thing? Well, let me illustrate it in an example. What's the space of lines through the origin in R^{3}? To keep track of these lines, draw a sphere around the origin. Each line through the origin intersects this sphere in two points. Either one point is in the northern hemisphere and the other is in the southern hemisphere, or both are on the equator. So we can keep track of all our lines using points on the northern hemisphere and the equator, but identifying antipodal points on the equator. This is just the same as taking the closed 2dimensional ball and identifying antipodal points on the boundary! QED. The same argument works in higher dimensions too.
Now that we know a point in RP^{n} is just a line through the origin in R^{n+1}, it's easy to put coordinates on RP^{n}. There's one line through the origin passing through any point in R^{n+1}, but if we multiply the coordinates (x_{1},...,x_{n+1}) of this point by any nonzero number we get the same line. Thus we can use a list of n+1 real numbers to describe a point in RP^{n}, with the proviso that we get the same point in RP^{n} if someone comes along and multiplies them all by some nonzero number! These are called "homogeneous coordinates".
If you don't like the ambiguity of homogeneous coordinates, you can go right ahead and divide all the coordinates by the real number x_{1}, getting
(1, x_{2}/x_{1}, ... , x_{n+1}/x_{1})
which lets us describe a point in RP^{n} by n real numbers, as befits an ndimensional real manifold. Of course, this won't work if x_{1} happens to be zero! But we can divide by x_{2} if x_{2} is nonzero, and so on. One of them has to be nonzero, so we can cover RP^{n} with n+1 different coordinate patches corresponding to the regions where different x_{i}'s are nonzero. It's easy to change coordinates, too.
This makes everything very algebraic, which makes it easy to generalize RP^{n} by replacing the real numbers with other number systems. For example, to define "complex projective nspace" or CP^{n}, just replace the word "real" by the word "complex" in the last two paragraphs, and replace "R" by "C". CP^{n} is even more of a geometer's paradise than RP^{n}, because when you work with complex numbers you can solve all polynomial equations. Also, now there's no big difference between an ellipse and a hyperbola! This sort of thing is why CP^{n} is so widely used as a context for "algebraic geometry".
We can go even further and replace the real numbers by the quaternions, H, defining the "quaternionic projective nspace" HP^{n}. If we are careful about writing things in the right order, it's no problem that the quaternions are noncommutative... we can still divide by any nonzero quaternion, so we can cover HP^{n} with n+1 different coordinate charts and freely change coordinates as desired.
We can try to go even further and use the octonions, O. Can we define "octonionic projective nspace", OP^{n}? Well, now things get tricky! Remember, the octonions are nonassociative. There's no problem defining OP^{1}; we can cover it with two coordinate charts, corresponding to homogeneous coordinates of the form
(x, 1)
and
(1, y),
and we can change coordinates back and forth with no problem. This amounts to taking O and adding a single point at infinity, getting the 8dimensional sphere S^{8}. This is part of a pattern:
RP^{1} = S^{1}
CP^{1} = S^{2}
HP^{1} = S^{4}
OP^{1} = S^{8}
I discussed the implications of this pattern for Bott periodicity in "week105".
We can also define OP^{2}. Here we have 3 coordinate charts corresponding to homogeneous coordinates of the form
(1, y, z),
(x, 1, z),
and
(x, y, 1).
We can change back and forth between coordinate systems, but now we have to check that if we start with the first coordinate system, change to the second coordinate system, and then change back to the first, we wind up where we started! This is not obvious, since multiplication is not associative. But it works, thanks to a couple of identities that are not automatic in the nonassociative context, but hold for the octonions:
(xy)^{1} = y^{1} x^{1}
and
(xy)y^{1} = x.
Checking these equations is a good exercise for anyone who wants to understand the octonions.
Now for the cool part: OP^{2} is where it ends!
We can't define OP^{n} for n greater than 2, because the nonassociativity keeps us from being able to change coordinates a bunch of times and get back where we started! You might hope that we could weasel out of this somehow, but it seems that there is a real sense in which the higherdimensional octonionic projective spaces don't exist.
So we have a fascinating situation: an infinite tower of RP^{n}'s, an infinite tower of CP^{n}'s, an infinite tower of HP^{n}'s, but an abortive tower of OP^{n}'s going only up to n = 2 and then fizzling out. This means that while all sorts of geometry and group theory relating to the reals, complexes and quaternions fits into infinite systematic patterns, the geometry and group theory relating to the octonions is quirky and mysterious.
We often associate mathematics with "classical" beauty, patterns continuing ad infinitum with the ineluctable logic of a composition by some divine Bach. But when we study OP^{2} and its implications, we see that mathematics also has room for "exceptional" beauty, beauty that flares into being and quickly subsides into silence like a piece by Webern. Are the fundamental laws of physics based on "classical" mathematics or "exceptional" mathematics? Since our universe seems unique and special  don't ask me how would we know if it weren't  Witten has suggested the latter. Indeed, it crops up a lot in string theory. This is why I'm trying to learn about the octonions: a lot of exceptional objects in mathematics are tied to them.
I already discussed this a bit in "week64", where I sketched how there are 3 infinite sequences of "classical" simple Lie groups corresponding to rotations in R^{n}, C^{n}, and H^{n}, and 5 "exceptional" simple Lie groups related to the octonions. After studying it all a bit more, I can now go into some more detail.
In order of increasing dimension, the 5 exceptional Lie groups are called G2, F4, E6, E7, and E8. The smallest, G2, is easy to understand in terms of the octonions: it's just the group of symmetries of the octonions as an algebra. It's a marvelous fact that all the bigger ones are related to OP^{2}. This was discovered by people like Freudenthal and Tits and Vinberg, but a great place to read about it is the following fascinating book:
1) Boris Rosenfeld, Geometry of Lie Groups, Kluwer Academic Publishers, 1997.
The space OP^{2} has a natural metric on it, which allows us to measure distances between points. This allows us to define a certain symmetry group OP^{2}, the group of all its "isometries", which are transformations preserving the metric. This symmetry group is F4!
However, there is another bigger symmetry group of OP^{2}. As in real projective nspace, the notion of a "line" makes sense in OP^{2}. One has to be careful: these are octonionic "lines", which have 8 real dimensions. Nonetheless, this lets us define the group of all "collineations" of OP^{2}, that is, transformations that take lines to lines. This symmetry group is E6! (Technically speaking, this is a "noncompact real form" of E6; the rest of the time I'll be talking about compact real forms.)
To get up to E7 and E8, we need to take a different viewpoint, which also gives us another way to get E6. The key here is that the tensor product of two algebras is an algebra, so we can tensor the octonions with R, C, H, or O and get various algebras:
I'm not making this up: it's all in Rosenfeld's book! The poet Lisa Raphals suggested calling the octooctonions the "highoctane octonions", which I sort of like. But compared to Rosenfeld, I'm a model of restraint: I won't even mention the dyoctonions, duoctonions, split octonions, semioctonions, split semioctonions, 1/4octonions or 1/8octonions  for the definitions of these, you'll have to read his book.
Apparently one can define projective planes for all of these algebras, and all these projective planes have natural metrics on them, all of them same general form. So each of these projective planes has a group of isometries. And, lo and behold:
Now I still don't understand this as well as I'd like to  I'm not sure how to define projective planes for all these algebras (though I have my guesses), and Rosenfeld is unfortunately a tad reticent on this issue. But it looks like a cool way to systematize the study of the expectional groups! That's what I want: a systematic theory of exceptions.
I want to say a bit more about the above, but first let me note that there are lots of other ways of thinking about the exceptional groups. A great source of information about them is the following posthumously published book by the great topologist Adams:
2) John Frank Adams, Lectures on Exceptional Lie Groups, eds. Zafer Mahmud and Mamoru Mimura, University of Chicago Press, Chicago, 1996.
He has a bit about octonionic constructions of G2 and F4, but mostly he concentrates on constructions of the exceptional groups using classical groups and spinors.
In "week90" I explained Kostant's constructions of F4 and E8 using spinors in 8 dimensions and triality  which, as noted in "week61", is just another way of talking about the octonions. Unfortunately I don't yet see quite how this relates to the above stuff, nor do I see how to get E6 and E7 in a beautiful way using Kostant's setup.
There's also a neat construction of E8 using spinors in 16 dimensions! Adams gives a nice explanation of this, and it's also discussed in the classic tome on string theory:
3) Michael B. Green, John H. Schwarz, and Edward Witten, Superstring Theory, two volumes, Cambridge U. Press, Cambridge, 1987.
The idea here is to take the direct sum of the Lie algebra so(16) and its 16dimensional lefthanded spinor representation S+ to get the Lie algebra of E8. The bracket of two guys in so(16) is defined as usual, the bracket of a guy in so(16) and a guy in S+ is defined to be the result of acting on the latter by the former, and the bracket of two guys in S+ is defined to be a guy in S+ by dualizing the map
so(16) ⊗ S+ → S+
to get a map
S+ ⊗ S+ → so(16).
This is a complete description of the Lie algebra of E8!
Anyway, there are lots of different ways of thinking about exceptional groups, and a challenge for the octonionic approach is to systematize all these ways.
Now I want to wrap up by saying a bit about how the exceptional Jordan algebra fits into the above story. Jordan algebras were invented as a way to study the selfadjoint operators on a Hilbert space, which represent observables in quantum mechanics. If you multiply two selfadjoint operators A and B the result needn't be selfadjoint, but the "Jordan product"
A o B = (AB + BA)/2
is selfadjoint. This suggests seeing what identities the Jordan product satisfies, cooking up a general notion of "Jordan algebra", seeing how much quantum mechanics you can do with an arbitrary Jordan algebra of observables, and classifying Jordan algebras if possible.
We can define a "projection" in a Jordan algebra to be an element A with A o A = A. If our Jordan algebra consists of selfadjoint operators on the complex Hilbert space C^{n}, a projection is a selfadjoint operator whose only eigenvalues are zero and one. Physically speaking, this corresponds to a "yesorno question" about our quantum system. Geometrically speaking, such an operator is a projection onto some subspace of our Hilbert space. All this stuff also works if we start with the real Hilbert space R^{n} or the quaternionic Hilbert space H^{n}.
In these special cases, one can define a "minimal projection" to be a projection on a 1dimensional subspace of our Hilbert space. Physically, minimal projections correspond to "pure states"  states of affairs in which the answer to some maximally informative question is "yes", like "is the z component of the angular momentum of this spin1/2 particle equal to 1/2?" Geometrically, the space of minimal projections is just the space of "lines" in our Hilbert space. This is either RP^{n1}, or CP^{n1}, or HP^{n1}, depending on whether we're working with the reals, complexes or quaternions. So: the space of pure states of this sort of quantum system is also a projective space! The relation between quantum theory and "projective geometry" has been intensively explored for many years. You can read about it in:
4) V. S. Varadarajan, Geometry of Quantum Theory, SpringerVerlag, Berlin, 2nd ed., 1985.
Most people do quantum mechanics with complex Hilbert spaces. Real Hilbert spaces are apparently too boring, but some people have considered the quaternionic case:
5) Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford U. Press, Oxford, 1995.
If our Hilbert space is the complex Hilbert space C^{n}, its group of symmetries is usually thought of as U(n)  the group of n×n unitary matrices. This group also acts as symmetries on the Jordan algebra of selfadjoint n×n complex matrices, and also on the space CP^{n1}.
Similarly, if we start with R^{n}, we get the group of orthogonal n×n matrices O(n), which acts on the Jordan algebra of real selfadjoint n×n matrices and on RP^{n1}.
Likewise, if we start with H^{n}, we get the group Sp(n), which acts on the Jordan algebra of quaternionic selfadjoint n×n matrices and on HP^{n1}.
This pretty much explains how the classical groups are related to different flavors of quantum mechanics.
Now what about the octonions? Well, here we can only go up to n = 3, basically for the reasons explained before: the same stuff that keeps us from defining octonionic projective spaces past a certain point keeps us from getting Jordan algebras! The interesting case is the Jordan algebra of 3×3 selfadjoint octonionic matrices. This is called the "exceptional Jordan algebra", J. The group of symmetries of this is  you guessed it, F4. One can also define a "minimal projection" in J and the space of these is OP^{2}.
Is it possible that octonionic quantum mechanics plays some role in physics?
I don't know.
Anyway, here is my hunch about the bioctonionic, quateroctonionic, and octooctonionic projective planes. I think to define them you should probably tensor the exceptional Jordan algebra with C, H, and O, respectively, and take the space of minimal projections in the resulting algebra. Rosenfeld seems to suggest this is the way to go. However, I'm vague about some important details, and it bugs me, because the special identities I needed above to define OP^{2} are related to O being an alternative algebra, but C ⊗ O, H ⊗ O and O ⊗ O are not alternative.
I should add that in addition to octonionic projective geometry, one can do octonionic hyperbolic geometry. One can read about this in Rosenfeld and also in the following:
6) Daniel Allcock, Reflection groups on the octave hyperbolic plane, University of Utah Mathematics Department preprint.
Addenda: Here's an email from David Broadhurst, followed by various remarks.
John:The theorem that fails for quaternions but holds for R and C is the "Pappus theorem", discussed in "week145".Shortly before his death I spent a charming afternoon with Paul Dirac. Contrary to his reputation, he was most forthcoming.
Among many things, I recall this: Dirac explained that while trained as an engineer and known as a physicist, his aesthetics were mathematical. He said (as I can best recall, nearly 20 years on): At a young age, I fell in love with projective geometry. I always wanted to use to use it in physics, but never found a place for it.
Then someone told him that the difference between complex and quaternionic QM had been characterized as the failure of theorem in classical projective geometry.
Dirac's face beamed a lovely smile: Ah he said, it was just such a thing that I hoped to do.
I was reminded of this when bactracking to your "week106", today.
Best,
David
Next, a bit about OP^{n}. There are different senses in which we can't define OP^{n} for n greater than 2. One is that if we try to define coordinates on OP^{n} in a similar way to how we did it for OP^{2}, nonassociativity keeps us from being able to change coordinates a bunch of times and get back where we started! It's definitely enlightening to see how the desired transition functions g_{ij} fail to satisfy the necessary cocycle condition g_{ij} g_{jk} = g_{ik} when we get up to OP^{3}, which would require 4 charts.
But, a deeper way to think about this emerged in conversations I've had with James Dolan. Stasheff invented a notion of "A_{∞} space", which is a pointed topological space with a product that is associative up to homotopy which satisfies the pentagon identity up to... etc. Any A_{∞} space G has a classifying space BG such that
Ω(BG) ~ G.
In other words, BG is a pointed space such that the space of loops based at this point is homotopy equivalent to G. One can form this space BG by the Milnor construction: sticking in one 0simplex, one 1simplex for every point of G, one 2simplex for every triple (g,h,k) with gh = k, one 3simplex for every associator, and so on. If we do this where G is the group of lengthone elements of R (i.e. Z/2) we get RP^{∞}, as we expect, since
RP^{∞} = B(Z/2).
Even better, at the nth stage of the Milnor construction we get a space homeomorphic to RP^{n}. Similarly, if we do this where G is the group of lengthone elements of C or H we get CP^{∞} or HP^{∞}. But if we take G to be the units of O, which has a product but is not even homotopyassociative, we get OP^{1} = S^{7} at the first step, OP^{2} at the second step, ... but there's no way to perform the third step!
Next: here's a little more information on the octonionic, bioctonionic, quateroctonionic and octooctonionic projective planes. Rosenfeld claims that the groups of isometries of these planes are F4, E6, E7, and E8, respectively. The problem is, I can't quite understand how he constructs these spaces, except for the plain octonionic case.
It appears that these spaces can also be constructed using the ideas in Adams' book. Here's how it goes.
According to:
6) Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp. 313316.
the above spaces are the octonionic, bioctonionic, quateroctonionic and octooctonionic projective planes, respectively. However, I don't yet fully understand the connection.
I thank Tony Smith for pointing out the reference to Besse (who, by the way, is apparently a cousin of the famous Bourbaki). Thanks also go to Allen Knutson for showing me a trick for finding the maximal rank subalgebras of a simple Lie algebra.
Next, here's some more stuff about the biquaternions, bioctonions, quaterquaternions, quateroctonions and octooctonions! I wrote this extra stuff as part of a post to sci.physics.research on November 8, 1999....
One reason people like these algebras is that some of them  the associative ones  are also Clifford algebras. I talked a bit about Clifford algebras in "week105", but just remember that we define the Clifford algebra C_{p,q} to be the associative algebra you get by taking the real numbers and throwing in p square roots of 1 and q square roots of 1, all of which anticommute with each other. This algebra is very important for understanding spinors in spacetimes with p space and q time dimensions. (It's also good for studying things in other dimensions, so things can get a bit tricky, but I don't want to talk about that now.)For example: if you just thrown in one square root of 1 and no square roots of 1, you get C_{1,0}  the complex numbers!
Similarly, one reason people like the quaternions is because they are C_{2,0}. Start with the real numbers, throw in two square roots of 1 called I and J, make sure they anticommute (IJ = JI) and voila  you've got the quaternions!
Similarly, one reason people like the biquaternions is because they are C_{2,1}. You take the quaternions and complexify them  this amounts to throwing in an extra number i that's a square root of 1 and commutes with the quaternionic I and J  and you get an algebra which is also generated by I, J, and K = iI. Note that I, J, and K all anticommute, and K is a square root of 1. Thus the biquaternions are C_{2,1}!
Similarly, one reason people like the quaterquaternions is because they are C_{2,2}. You take the quaternions and quaternionify them  this amounts to throwing in two square roots of 1, say i and j, which anticommute but which commute with the quaternionic I and J  and you get an algebra which is also generated by I, J, K = iI, and L = jI. Note that I, J, K, and L all anticommute, and K and L are square roots of 1. Thus the quaterquaternions are C_{2,2}!
Now, as soon as we thrown the octonions into the mix we don't get Clifford algebras anymore, since octonions aren't associative, while Clifford algebras are. However, there are still relationships to Clifford algebras. For example, suppose we look at all the linear transformations of the octonions generated by the left multiplication operations
x → ax
This is an associative algebra, and it turns out to be all linear transformations of the octonions, regarded as an 8dimensional real vector space. In short, it's just the algebra of 8x8 real matrices. And this is C_{6,0}.
If you do the same trick for the bioctonions, quateroctonions and octooctonions, you get other Clifford algebras... but I'll leave the question of which ones as a puzzle for the reader. If you need some help, look at the "Footnote" in "week105".
Perhaps the fanciest example of this trick concerns the biquateroctonions. Now actually, I've never heard anyone use this term for the algebra C ⊗ H ⊗ O! The main person interested in this algebra is Geoffrey Dixon, and he just calls it T. But anyway, if we look at the algebra of linear transformations of C ⊗ H ⊗ O generated by left multiplications, we get something isomorphic to the algebra of 16 x 16 complex matrices. And this in turn is isomorphic to C_{9,0}.
The biquateroctonions play an important role in Dixon's grand unified theory of the electromagnetic, weak and strong forces. There are lots of nice things about this theory  for example, it gets the right relationships between weak isospin and hypercharge for the fermions in any one generation of the Standard Model (though, as in the Standard Model, the existence of 3 generations needs to be put in "by hand"). It may or may not be right, but at least it comes within shooting distance!
You can read a bit more about his work in "week59".
© 1997 John Baez
baez@math.removethis.ucr.andthis.edu
