|
|
|
|
When I write This Week's Finds as rarely as I do these days, so much stuff builds up that I completely despair of ever getting to all of it... so I'll just randomly mention a few cool things that are on the top of my mind right now.
First of all, here's a great new review article on spin foams. If you're trying to understand spin foam models of quantum gravity, this is the place to start:
1) Daniele Oriti, Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity, Rep. Prog. Phys. 64 (2001), 1489-1544. Also available at gr-qc/0106091.
You'll learn how spin foam models naturally show up in all sorts of different approaches to quantum gravity: loop quantization, path integral approaches, lattice field theory, matrix models, and category-theoretic approaches.
Secondly, here's a great introduction to n-categories and topology:
2) Tom Leinster, Topology and higher-dimensional category theory: the rough idea, available at math.CT/0106240.
As he says, this is a "Friday-afternoonish description of some of the dreams people have for higher-dimensional category theory and its interactions with topology". Much more readable than the Monday-morningish papers where people put in all the details!
And next, here is some stuff I have been thinking about lately.
As you're probably sick to death of hearing, I'm interested in category theory and also normed division algebras: the real numbers, complex numbers, quaternions and octonions. There's no instantly obvious relationship between these topics, but naturally I've tried to find one, since this would let me unify two of my obsessions into one big super-obsession. I recently made a bunch of progress, thanks to finding these papers:
3) Markus Rost, On the dimension of a composition algebra, Documenta Mathematica 1 (1996), 209-214. Available at http://www.mathematik.uni-bielefeld.de/DMV-J/vol-01/10.html
4) Dominik Boos, Ein tensorkategorieller Zugang zum Satz von Hurwitz (A tensor-categorical approach to Hurwitz's theorem), Diplomarbeit ETH Zurich, March 1998, available at http://www.mathematik.uni-bielefeld.de/~rost/data/boos.pdf
I'd like to explain what the problem is and how these papers solve it.
Part of the fun of category theory is that it lets you take mathematical arguments and generalize them to their full extent by finding the proper context for them: that is, by figuring out in exactly what sort of category you can carry out the argument. Out of laziness and ignorance, people usually work in the category of sets as a kind of "default setting". This category has many wonderful features - it's like a machine that chops, slices, dices, grates, liquefies and purees - but usually you don't need all these features to carry out a particular task. So, one job of a category theorist is to figure out what features are actually needed in a given situation, and isolate the kind of category that has those features.
A "kind of category" is sometimes called a "doctrine". I believe this term was invented by Lawvere. It must have some technical definition, but luckily I don't know it, so I will not be restrained by it here. I'll just talk in a sloppy way about this question: "in what doctrine can we define the concept of a normed division algebra?" It'll get technical for a while, so most of you may want to leave, but then some pretty pictures will show up, so make sure to come back then.
First think a minute about "algebras". Here by an "algebra" I mean a finite-dimensional real vector space with a not-necessarily-associative bilinear product and an element that's both the left and right unit for this product. We can define algebras like this using the category Vect consisting of real vector spaces and linear operators, without resorting to full power of the category of sets - as long as we use the tensor product in Vect. We start by saying an algebra is an object A in Vect together with a product
m: A tensor A → A
and unit
i: I → A
where I is the unit object for the tensor product - that is, the real numbers. In case you're confused: the map i here is just the linear operator sending the real number 1 to the unit element of A; we're using a standard trick for expressing elements as maps. Given this stuff, we can write the left and right unit laws by saying this diagram commutes:
I tensor A <--------- A -------> A tensor I
| | |
i tensor 1_A | |1_A | 1_A tensor i
| | |
V m V m V
A tensor A ---------> A <-------- A tensor A
where the unlabelled arrows are some obvious isomorphisms coming from
the fact that I is the unit for the tensor product.
Now, this definition could have been stated in any category with tensor products; or more technically, any "monoidal category". So the right doctrine for talking about algebras of this sort is the doctrine of monoidal categories.
What's the right doctrine for defining associative algebras? Well, we can write down another commutative diagram to state the associative law:
(A tensor A) tensor A ----------------> A tensor (A tensor A)
| |
m tensor 1_A | | 1_A tensor m
| |
V V
m m
A tensor A -----------> A <----------- A tensor A
where again the unlabelled arrow is the obvious isomorphism. This works
fine in any monoidal category, so the right doctrine is again that of
monoidal categories. But instead of speaking of an "associative
algebra" in a monoidal category, folks usually call a gadget of
this sort a "monoid object"- see "week89" for more on this. The reason is that if
we take our monoidal category to be Set, a monoid object boils down to a
"monoid": a set with an associative product and unit element.
Lots of people like groups more than monoids. What's the right doctrine for defining groups? This time it's definitely NOT the doctrine of monoidal categories. The reason is that the equational laws satisfied by inverses in a group:
g g-1 = 1
g-1 g = 1
have duplicated and deleted arguments - the "g" shows up twice on the left side and not at all on the right! This is different from the associative law
g (h k) = (g h) k
where each argument shows up once on each side of the equation.
In a monoidal category we can't "duplicate" or "delete" arguments: if X is an object in a monoidal category, there's no god-given map from X to X tensor X, or from X to 1. This means we can't use commutative diagrams in a monoidal category to express equational laws that duplicate or delete arguments.
However, we can duplicate and delete arguments if we're in a "category with finite products" - a nice sort of monoidal category where we do have maps from X to X tensor X and from X to 1. The best example of this is the category of sets, where the "tensor product" is just the usual Cartesian product. This is why we can easily define groups in the category of sets! More generally, we can define "group objects" in any category with finite products.
So, the right doctrine for talking about groups - or more precisely, group objects - is the doctrine of categories with finite products.
By the way, if you think this stuff is too abstract to be useful, take a peek at "week54" and "week115", where I described how group objects show up in algebraic topology. But beware: back then I was engaging in a bit of overkill, and working in the doctrine of "categories with finite limits". This more powerful doctrine also lets you define gadgets with partially defined operations, like "category objects". But for group objects, finite products are all we really need.
Gradually getting to the point, let us now ask: what's the right doctrine for talking about division algebras? It's definitely not the doctrine of monoidal categories. It's not even the doctrine of categories with finite products! The problem is that a division algebra is defined to be an algebra such that xy = 0 implies x = 0 or y = 0. This condition is not even an equational law: it doesn't say some equation holds, it says "this equation implies this one or that one". To express such fancier conditions as commutative diagrams, we need a more powerful doctrine.
I'm too lazy to figure out exactly what we need, but certainly the doctrine of "topoi" will do. If you don't know what a topos is, give yourself 40 lashes and read this:
5) John Baez, Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html
However, there are lots of reasons to avoid working in such a powerful doctrine - basically, it greatly limits the generality with which one can discuss a subject.
So it's very interesting to see how much better we can do if we're interesed in normed division algebras. These are algebras equipped with a norm such that
|xy| = |x| |y|
and if we're working in the category of real vector spaces, the only examples are the real numbers, the complex numbers, the quaternions and the octonions. These have all sorts of important applications in physics, so it's good to see what doctrine we need to talk about them.
The problem is that the norm is nothing like a linear map. To get around this, it's better to work with the inner product, which is related to the norm by
|x|2 = <x,x>
The inner product is bilinear, so have a chance of talking about it in the doctrine of monoidal categories. Unfortunately, there are a couple of problems:
First of all, it's tough to state the positive definiteness of the inner product:
if x is nonzero, then <x,x> is greater than 0.
The easiest way around this is to relax a bit, and instead of demanding that our algebra have an inner product <x,y>, simply demand that it have a nondegenerate bilinear form g(x,y). Believe it or not, this condition can be stated in any monoidal category. It's easiest to do this using pictures - not commutative diagrams, but an equivalent approach using pictures that look a bit like Feynman diagrams. These days, lots of mathematical physicists use pictures like this to do calculations in monoidal categories. There are lots of places to learn this stuff, but if you want something online, it's easiest for me to point you to my notes on quantum gravity:
6) John Baez, Toby Bartels, and Miguel Carrion, Quantum gravity seminar, http://math.ucr.edu/home/baez/qg.html
Okay. Now that you've read those notes, you know what to do! We assume our algebra A is equipped with maps
g: A tensor A → I
and
h: I → A tensor A
which we draw as
| |
| |
\ /
\ /
\_______/
g
and
___h___
/ \
/ \
/ \
| |
| |
respectively. We demand that
| | |
| | |
| | |
| __h__ | __h__ |
| / \ = | = / \ |
| / \ | / \ |
| / | | | \ |
\ / | | | \ /
\_____/ | | | \_____/
g | | | g
| | |
| | |
| | |
which says that the bilinear form g is nondegenerate. To get
further, we'll also demand that
| | | |
| | | |
\ / | |
\ / | |
\ / = | |
/ | |
/ \ | |
/ \ | |
| | \ /
\ / \ /
\_/ \_____/
g g
This says that the bilinear form g is symmetric, that is:
g(x,y) = g(y,x).
But we can only state this equation if we're in a monoidal category where we can "switch arguments", which in pictures goes like this:
\ /
\ /
/
/ \
/ \
A monoidal category with this feature is called a "symmetric
monoidal category" (or more generally a "braided monoidal
category", but I don't want to get into those complications here).
So far, so good! The second problem is figuring out how to state the condition |xy| = |x| |y|. If we translate this into a condition on our bilinear form g, we get
g(xy,xy) = g(x,x) g(y,y)
An algebra with a nondegenerate bilinear form having this property is called a "composition algebra". Hurwitz showed that such an algebra must have dimension 1, 2, 4, or 8. However, there are examples other than the famous four, coming from bilinear forms g that aren't positive definite. For example, there are the "split quaternions" in dimension 4, or the "split octonions" in dimension 8.
Now, the problem with the above equational law is that it involves duplication of arguments. But we can get around this problem by a standard trick called "polarization", which people use a lot in quantum mechanics.
First let's polarize the argument x. To do this, note that we have
g(xy,xy) = g(x,x) g(y,y)
g(x'y,x'y) = g(x',x') g(y,y)
and also
g((x+x')y,(x+x')y) = g(x+x',x+x') g(y,y).
Subtracting the first two equations from the last and then dividing by 2, we get
g(xy,x'y) = g(x,x') g(y,y).
See? We've eliminated the duplication of the argument x. This new equation obviously implies the original one.
Next we polarize the argument y. We have
g(xy,x'y) = g(x,x') g(y,y)
g(xy',x'y') = g(x,x') g(y',y')
and also
g(x(y+y'),x'(y+y')) = g(x,x') g(y+y',y+y').
Subtracting the first two equations from the last one, we get
g(xy,x'y') + g(xy',x'y) = 2 g(x,x') g(y,y')
Now there is no duplication of arguments. We've paid a price, though: now our equation involves addition, so we can only write it down if our category has the extra feature that we can add morphisms. For this, we want our category to be "additive".
So: the right doctrine in which to define composition algebras is the doctrine of symmetric monoidal additive categories!
(Technical note: here we want the monoidal and additive structures to get along nicely: tensoring of morphisms should be bilinear.)
Let me summarize by giving all the details. A "composition object" is an object A in a symmetric monoidal additive category which is equipped with morphisms
\ / A tensor A
\ /
\ / |
\ / |
m m | "multiplication"
| |
| V
|
| A
I
i
| |
| i | "unit"
| |
| V
A
| | A tensor A
| |
| | |
\ / g | "bilinear form"
\_____/ V
g
I
I
__h__
/ \ |
/ \ h | "inverse bilinear form"
| | |
| | V
| |
A tensor A
satisfying the equations already shown:
| | |
| | |
| | |
| __h__ | __h__ |
| / \ = | = / \ |
| / \ | / \ |
| / | | | \ |
\ / | | | \ /
\_____/ | | | \_____/
g | | | g
| | |
| | |
| | |
and
| | | |
| | | |
\ / | |
\ / | |
\ / = | |
/ | |
/ \ | |
/ \ | |
| | \ /
\ / \ /
\_/ \_____/
g g
together with the left and right unit laws:
| | |
| | |
| | |
i | | | i
| | | | |
| | | | |
\ | | | /
\ / | \ /
\ / = | = \ /
m | m
| | |
| | |
| | |
| | |
| | |
and best of all, the equation
g(xy,x'y') + g(xy',x'y) = 2 g(x,x') g(y,y')
translated into pictures like this:
\ / \ / \ \ / / \ \ / /
\ / \ / \ \ / / \ \ / /
m m + \ m / = 2 \ / /
\ / \ | / \___/ \___/
\___/ \ / / g g
g / /
/ \ /
| m
\__/
g
Now, given all this stuff, we can define the "dimension" of our
composition algebra to be the value of this morphism from I to I:
_______
/ \
/ \
| |
| |
\ /
\_______/
g
This reduces to the usual dimension of the algebra A when we're in
the category Vect. Of course, only in certain categories is
this dimension bound to be a number - namely, those categories where
every morphism from I to I is some number times the identity morphism.
By making an extra assumption like this, Boos is able to give a "picture proof" that in a large class of symmetric monoidal additive categories, every composition object has dimension 1, 2, 4 or 8. This is great, because it means we can talk about things like real, complex, quaternionic and octonionic objects in a wide variety of categories! He doesn't prove such objects exist, but I think this should be easy, at least with some extra assumptions which would allow us to construct them "by hand", mimicking standard constructions of the normed division algebras.
But now I must warn you of some things. Boos doesn't state his result the way I would! Instead of working with "composition objects" (which appear to be my own invention), he works with "vector product algebras". These are modelled, not after the normed division algebras themselves, but after their "imaginary parts". These have both an inner product and a "vector product".
For example, the imaginary quaternions form a 3-dimensional vector product algebra with vector product given by
a x b = (1/2)(ab - ba).
This is just the usual cross product! The same formula makes the imaginary octonions into a 7-dimensional vector product algebra, the imaginary complex numbers into a boring 1-dimensional one... and the imaginary real numbers into an even more boring 0-dimensional one.
Boos writes down the axioms for a vector product algebra using pictures much like I just did for a composition object, and he shows that under some pretty mild conditions you can freely go back and forth between the two concepts.
I think you can summarize his theorem on vector product algebras as follows: in all symmetric monoidal R-linear categories where R is a commutative ring containing Z[1/2] and I is a simple object, vector product algebras must have dimension 0, 1, 3, or 7. He doesn't state his result quite this way, but I'm pretty sure that's what it boils down to. As for the jargon: a category is "R-linear" if the homsets are R-modules and composition of morphisms is bilinear; for monoidal categories we also want tensoring morphisms to be bilinear. The ring Z[1/2] consists of all fractions with a power of 2 in the denominator - Boos needs this because he needs to divide by 2 at some point in his argument. For an R-linear category, an object I is "simple" if hom(I,I) = R. This allows us to interpret the dimension of our vector product algebra as an element of R - which Boos shows is actually one of the integers 0, 1, 3, or 7.
Let me conclude by showing you Boos' main axiom for vector product algebras, written in terms of pictures:
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \___/ | \ / \ / \____/ / \ + | = 2 / - | | - ____ / \ | / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \Ain't it cool? Fans of knot theory will be struck by the resemblance to various "skein relations". Fans of physics will be reminded of Feynman diagrams. But what is the secret inner meaning?
© 2001 John Baez
baez@math.removethis.ucr.andthis.edu
|
|
|
|