Also available at http://math.ucr.edu/home/baez/week106.html
July 23, 1997
This Week's Finds in Mathematical Physics - Week 106
John Baez
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 n-dimensional 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 well-known one is to
take R^n and add on one extra "point at infinity", obtaining the
n-dimensional 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
n-dimensional sphere but "projective n-space". 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 n-space is also called RP^n. The R is for "real", since
this is actually "real projective n-space". 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/sqrt(1 + |x|^2)
Here x stands for a vector in R^n and |x| is its length. Dividing the
vector x by sqrt(1 + |x|^2) 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 n-space 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 n-space. 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 n-space!
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 n-dimensional 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 2-dimensional 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
n-dimensional 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 n-space" 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 n-space" 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 n-space", 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
8-dimensional 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 higher-dimensional 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 n-space, 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:
The algebra (R tensor O) is just the octonions.
The algebra (C tensor O) is called the "bioctonions".
The algebra (H tensor O) is called the "quateroctonions".
Finally, the algebra (O tensor O) is called the "octooctonions".
I'm not making this up: it's all in Rosenfeld's book! The poet Lisa
Raphals suggested calling the octooctonions the "high-octane
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, 1/4-octonions or
1/8-octonions - 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:
The group of isometries of the octonionic projective plane is F4.
The group of isometries of the bioctonionic projective plane is E6.
The group of isometries of the quateroctonionic projective plane is E7.
The group of isometries of the octooctonionic projective plane is E8.
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 16-dimensional left-handed spinor representation Spin+ 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 Spin+ is defined to
be the result of acting on the latter by the former, and the bracket
of two guys in Spin+ is defined to be a guy in Spin+ by dualizing the
map
so(16) x Spin+ -> Spin+
to get a map
Spin+ x Spin+ -> 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 self-adjoint operators on a Hilbert space, which
represent observables in quantum mechanics. If you multiply two
self-adjoint operators A and B the result needn't be self-adjoint, but
the "Jordan product"
A o B = (AB + BA)/2
is self-adjoint. 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 self-adjoint
operators on the complex Hilbert space C^n, a projection is a
self-adjoint operator whose only eigenvalues are zero and one.
Physically speaking, this corresponds to a "yes-or-no 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 1-dimensional 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 spin-1/2
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^{n-1}, or CP^{n-1}, or HP^{n-1}, 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, Springer-Verlag,
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 nxn unitary
matrices. This group also acts as symmetries on the Jordan algebra of
self-adjoint nxn complex matrices, and also on the space CP^{n-1}.
Similarly, if we start with R^n, we get the group of orthogonal nxn
matrices O(n), which acts on the Jordan algebra of real self-adjoint
nxn matrices and on RP^{n-1}. Likewise, if we start with H^n, we get
the group Sp(n), which acts on the Jordan algebra of quaternionic
self-adjoint nxn matrices and on HP^{n-1}. 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 x 3 self-adjoint 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 tensor O, H tensor O
and O tensor 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.
Quote of the week:
"Mainstream mathematics" is a name given to mathematics that more
fittingly belongs on Sunset Boulevard - Gian-Carlo Rota, Indiscrete Thoughts
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Addenda: Here's an email from David Broadhurst, followed by various
remarks:
John:
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
The theorem that fails for quaternions but holds for R and C is the
"Pappus theorem", discussed in "week145".
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_infinity 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_infinity space G has a
classifying space BG such that
Loops(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 0-simplex, one
1-simplex for every point of G, one 2-simplex for every triple (g,h,k)
with gh = k, one 3-simplex for every associator, and so on. If we do
this where G is the group of length-one elements of R (i.e. Z/2) we get
RP^infinity, as we expect, since
RP^infinity = 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 length-one elements of C or H we get CP^infinity or
HP^infinity. But if we take G to be the units of O, which has a
product but is not even homotopy-associative, 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.
The Lie algebra F4 has a subalgebra of maximal rank isomorphic to
so(9). The quotient space is 16-dimensional - twice the dimension
of the octonions. It follows that the Lie group F4 mod the subgroup
generated by this subalgebra is a 16-dimensional Riemannian manifold
on which F4 acts by isometries.
The Lie algebra E6 has a subalgebra of maximal rank isomorphic to
so(10) x u(1). The quotient space is 32-dimensional - twice the
dimension of the bioctonions. It follows that the Lie group E6 mod
the subgroup generated by this subalgegra is a 32-dimensional
Riemannian manifold on which E6 acts by isometries.
The Lie algebra E7 has a subalgebra of maximal rank isomorphic to
so(12) x su(2). The quotient space is 64-dimensional - twice the
dimension of the quateroctonions. It follows that the Lie group E6
mod the subgroup generated by this subalgegra is a 64-dimensional
Riemannian manifold on which E7 acts by isometries.
The Lie algebra E8 has a subalgebra of maximal rank isomorphic to so(16).
The quotient space is 128-dimensional - twice the dimension of
the octooctonions. It follows that the Lie group E6 mod the
subgroup generated by this subalgegra is a 128-dimensional Riemannian
manifold on which E8 acts by isometries.
According to:
6) Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp.
313-316.
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 8-dimensional 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 tensor H tensor 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 tensor H tensor
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".
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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