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First, the astronomy picture of the week:
1) NASA Photojournal, 'Victoria Crater' at Meridiani Planum, http://photojournal.jpl.nasa.gov/catalog/PIA08813
This is the crater that NASA's rover called Opportunity has been exploring. It's 800 meters across. I like this picture just because it's beautiful. It was taken by the High Resolution Imaging Science Experiment on NASA's Mars Reconnaissance Orbiter.
Now, on to business! I want to talk about this paper, which took over 2 years to write:
2) John Baez, Aristide Baratin, Laurent Freidel and Derek Wise, Representations of 2-groups on infinite-dimensional 2-vector spaces, available as arXiv:0812.4969.
We can dream up the notion of "2-vector space" by pondering this analogy chart:
numbers vector spaces addition direct sum multiplication tensor product 0 the 0-dimensional vector space 1 the 1-dimensional vector spaceJust as you can add and multiply numbers, you can add and multiply vector spaces - but people call these operations "direct sum" and "tensor product", to make them sound more intimidating. These new operations satisfy axioms similar to the old ones. However, what used to be equations like this:
x + y = y + x
now become isomorphisms like this:
X + Y ≅ Y + X.
This means we're "categorifying" the concepts of plus and times.
The unit for addition of vector spaces is the 0-dimensional vector space, and the unit for multiplication of vector spaces is the 1-dimensional vector space.
But here's the coolest part. Our chart is like a snake eating its own tail. The first entry of the first column matches the last entry of the second column! The set of all "numbers" is the same as "the 1-dimensional vector space". If by "numbers" we mean complex numbers, these are both just C.
This suggests continuing the chart with a third column, like this:
numbers (C) vector spaces (Vect) 2-vector spaces (2Vect) addition direct sum direct sum multiplication tensor product tensor product 0 C0 Vect0 1 C1 Vect1Here C0 is short for the 0-dimensional vector space, while C1 is short for the 1-dimensional vector space - in other words the complex numbers, C. Vect is the category of all vector spaces. So, whatever a "2-vector space" is, to make the chart nice we'd better have Vect be the 1-dimensional 2-vector space. We can emphasize this by calling it Vect1.
In fact, about 15 years ago Kapranov and Voevodsky invented a theory of 2-vector spaces that makes all this stuff work:
3) Mikhail Kapranov and Vladimir Voevodsky, 2-categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Proc. Sympos. Pure Math. 56, Part 2, AMS, Providence, RI, 1994, pp. 177-259.
They mainly considered finite-dimensional 2-vector spaces. Every finite-dimensional vector space is secretly just Cn, or at least something isomorphic to that. Similarly, every finite-dimensional 2-vector space is secretly just Vectn, or at least something equivalent to that.
(You see, when we categorify once, equality becomes isomorphism. When we do it again, isomorphism becomes "equivalence".)
What's Vectn, you ask? Well, what's Cn? It's the set where an element is an n-tuple of numbers:
(x1, ..., xn)
So, Vectn is the category where an object is an n-tuple of vector spaces:
(X1, ..., Xn)
It's all pathetically straightforward. Of course we also need to know what's a morphism in Vectn. What's a morphism from
(X1, ..., Xn)
to
(Y1, ..., Yn)?
It's just the obvious thing: an n-tuple of linear operators
(f1: X1 → Y1, ..., fn: Xn → Yn)
And we compose these in the obvious way, namely "componentwise".
This may seem like an exercise in abstract nonsense, extending formal patterns just for the fun of it. But in fact, 2-vector spaces are all over the place once you start looking. For example, take the category of representations of a finite group, or the category of vector bundles over a finite set. These are finite-dimensional 2-vector spaces!
Here I can't resist a more sophisticated digression, just to impress you. The whole theory of Fourier transforms for finite abelian groups categorifies nicely, using these examples. Any finite abelian group G has "Pontryagin dual" G* which is again a finite abelian group. I explained how this works back in "week273". The Fourier transform is a map from functions on G to functions on G*. So, it's a map between vector spaces. But, lurking behind this is a map between 2-vector spaces! It's a map from representations of G to vector bundles over G*.
You can safely ignore that last paragraph if you like. But if you want more details, try section 6.1 of this old paper:
4) John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189. Also available as arXiv:q-alg/9609018.
As you can see from the title, I was trying to go beyond 2-vector spaces and think about "2-Hilbert spaces". That's because in quantum physics, we use Hilbert spaces to describe physical systems. Recent work on physics suggests that we categorify this idea and study 2-Hilbert spaces, 3-Hilbert spaces and so on - see "week58" for details. In the above paper I defined and studied finite-dimensional 2-Hilbert spaces. But a lot of the gnarly fun details of Hilbert space theory show up only for infinite-dimensional Hilbert spaces - and we should expect the same for 2-Hilbert spaces.
After these old papers on 2-vector spaces and 2-Hilbert spaces, various people came along and improved the whole story. For example:
5) Martin Neuchl, Representation Theory of Hopf Categories, Ph.D. dissertation, University of Munich, 1997. Chapter 2: 2-dimensional linear algebra. Available at http://math.ucr.edu/home/baez/neuchl.ps
6) Josep Elgueta, A strict totally coordinatized version of Kapranov and Voevodsky 2-vector spaces, to appear in Math. Proc. Cambridge Phil. Soc. Also available as arXiv:math/0406475.
7) Bruce Bartlett, The geometry of unitary 2-representations of finite groups and their 2-characters, available as arXiv/0807.1329.
In the last of these, Bruce worked out how finite-dimensional 2-Hilbert spaces arise naturally in certain topological quantum field theories!
Just as we can study representations of groups on vector spaces, we can study representations of "2-groups" on 2-vector spaces:
8) Magnus Forrester-Barker, Representations of crossed modules and cat1-groups, Ph.D. thesis, Department of Mathematics, University of Wales, Bangor, 2004. Available at http://www.maths.bangor.ac.uk/research/ftp/theses/forrester-barker.pdf
9) John W. Barrett and Marco Mackaay, Categorical representations of categorical groups, Th. Appl. Cat. 16 (2006), 529-557. Also available as arXiv:math/0407463.
10) Josep Elgueta, Representation theory of 2-groups on finite dimensional 2-vector spaces, available as arXiv:math.CT/0408120.
A group is a category with one object, all of whose morphisms are invertible. Similarly, a 2-group is a 2-category with one object, all of whose morphisms and 2-morphisms are invertible. Just as we can define "Lie groups" to be groups where the group operations are smooth, we can define "Lie 2-groups" to be 2-groups where all the 2-group operations are smooth. Lie groups are wonderful things, so we can hope Lie 2-groups will be interesting too. There are already lots of examples known. You can see a bunch here:
11) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 12 (2004), 423-491. Available at http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html and as arXiv:math/0307200.
However, Barrett and Mackaay discovered something rather upsetting. While Lie groups have lots of interesting representations on finite-dimensional vector spaces, Lie 2-groups don't have many representations on finite-dimensional 2-vector spaces!
In fact, the problem already shows up for representations of plain old Lie groups on 2-vector spaces. A Lie group can be seen as a special sort of Lie 2-group, where the only 2-morphisms are identity morphisms.
The problem is that unlike a vector space, a 2-vector space has a unique basis - at least up to isomorphism. In Cn there's an obvious basis consisting of vectors like
(1,0,0,...)
(0,1,0,...)
(0,0,1,...)
and so on, but there are lots of other bases too. But in Vectn the only basis goes like this:
(C1,C0,C0,...)
(C0,C1,C0,...)
(C0,C0,C1,...)
Well, I'm exaggerating slightly: we could replace C1 here by any other 1-dimensional vector space, and C0 by any other 0-dimensional vector space. That would give other bases - but they'd still be isomorphic to the basis shown above.
So, if we have a group acting on a finite-dimensional 2-vector space, it can't do much more than permute the basis elements. So, any representation of a group on a finite-dimensional 2-vector space gives an action of this group as permutations of a finite set!
That's okay for finite groups, since these can act in interesting ways as permutations of finite sets. But it's no good for Lie groups. Lie groups are usually infinite: they're manifolds. So, they have lots of actions on manifolds, but not many actions on finite sets.
This suggests that to study representations of Lie groups (or more general Lie 2-groups) on 2-vector spaces, we should invent some notion of "infinite-dimensional 2-vector space", where the basis can be not a finite set but an infinite set - indeed, something more like a manifold.
Luckily, such a concept was already lurking in the mathematical literature!
In the categorification game, it's always good when the concepts you invent shed light on existing issues in mathematics. And it's especially fun when you categorify a concept and get a concept that turns out to have been known - or at least partially known - under some other name. Then you're not just making up new stuff: you're seeing that existing math already had categorification built into it! This happens surprisingly often. That's why I take categorification so seriously.
The concept I'm talking about here is called a "field of Hilbert spaces". Roughly speaking, the idea is that you pick a set X, possibly infinite. X could be the real line, for example. Then a "field of Hilbert spaces" assigns to each point x in X a Hilbert space Hx.
As I've just described it, a measurable field of Hilbert spaces is an object in what we might call HilbX - a hairier, scarier relative of the tame little Vectn that I've been talking about.
Let's think about how HilbX differs from Vectn. First, the the finite number n has been replaced by an infinite set X. That's why HilbX deserves to be thought of as an infinite-dimensional 2-vector space.
Second, Vect has been replaced by Hilb - the category of Hilbert spaces. This suggests that HilbX is something more than a mere infinite-dimensional 2-vector space. It's closer to an infinite-dimensional 2-Hilbert space! So, we've departed somewhat from our original goal of inventing a notion of infinite-dimensional vector space. But that's okay, especially if we're interested in applications to quantum physics that involve analysis.
And here I must admit that I've left out some important details. When studying fields of Hilbert spaces, people usually bring in some analysis to keep the Hilbert space Hx from jumping around too wildly as x varies. They restrict attention to "measurable" fields of Hilbert spaces. To do this, they assume X is a "measurable space": a space with a sigma-algebra of subsets, like the Borel sets of the real line. Then they assume Hx depends in a measurable way on x.
The last assumption must be made precise. I won't do that here - you can see the details in our paper. But, here's an example of what I mean. Take X and partition it into countably many disjoint measurable subsets. For each one of these subsets, pick some Hilbert space H and let Hx = H for points x in that subset. So, the dimension of the Hilbert space Hx can change as x moves around, but only in a "measurable way". In fact, every measurable field of Hilbert spaces is isomorphic to one of this form.
So, a measurable field of Hilbert spaces on X is like a vector bundle over X, except the fibers are Hilbert spaces and there's no smoothness or continuity - the dimension of the fiber can "jump" in a measurable way.
If you've studied algebraic geometry, this should remind you of a "coherent sheaf". That's another generalization of a vector bundle that allows the dimension of the fiber to jump - but in an algebraic way, rather than a measurable way. One reason algebraic geometers like categories of coherent sheaves is because they need a notion of infinite-dimensional 2-vector space. Similarly, one reason analysts like measurable fields of Hilbert spaces is because they want their own notion of infinite-dimensional 2-vector space. Of course, they don't know this - if you ask, they'll strenuously deny it.
We learned most of what we know about measurable fields of Hilbert spaces from this classic book:
12) Jacques Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981.
This book was also helpful:
13) William Arveson, An Invitation to C*-Algebra, Chapter 2.2, Springer, Berlin, 1976.
As you might guess from the titles of these books, measurable fields of Hilbert spaces show up when we study representations of operator algebras that arise in quantum theory. For example, any commutative von Neumann algebra A is isomorphic to the algebra L∞(X) for some measure space X, and every representation of A comes from a measurable field of Hilbert spaces on X.
The following treatment is less detailed, but it explains how measurable fields of Hilbert spaces show up in group representation theory:
14) George W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, Benjamin-Cummings, New York, 1978.
I'll say a lot more about this at the very end of this post, but here's a quick, rough summary. Any sufficiently nice topological group G has a "dual": a measure space G* whose points are irreducible representations of G. You can build any representation of G from a measurable field of Hilbert spaces on G* together with a measure on G*. You build the representation by taking a "direct integral" of Hilbert spaces over G*. This is a generalization of writing a representation as a direct sum of irreducible representation. Direct integrals generalize direct sums - just as integrals generalize sums!
By the way, Mackey calls measurable fields of Hilbert spaces "measurable Hilbert space bundles". Those who like vector bundles will enjoy his outlook.
But let's get back to our main theme: representations of 2-groups on infinite-dimensional 2-vector spaces.
We don't know the general definition of an infinite-dimensional 2-vector space. However, for any measurable space X, we can define measurable fields of Hilbert spaces on X. We can also define maps between them, so we get a category, called Meas(X). Crane and Yetter call these "measurable categories".
I believe someday we'll see that measurable categories are a halfway house between infinite-dimensional 2-vector spaces and infinite-dimensional 2-Hilbert spaces. In fact, when we move up to n-vector spaces, it seems there could be n+1 different levels of "Hilbertness".
The conclusions of our paper include a proposed definition of 2-Hilbert space that can handle the infinite-dimensional case. So, why work with measurable categories? One reason is that they're they're well understood, thanks in part to the work of Dixmier - but also thanks to Crane and Yetter:
15) David Yetter, Measurable categories, Appl. Cat. Str. 13 (2005), 469-500. Also available as arXiv:math/0309185.
16) Louis Crane and David N. Yetter, Measurable categories and 2-groups, Appl. Cat. Str. 13 (2005), 501-516. Also available as arXiv:math/0305176.
The paper by Crane and Yetter studies representations of discrete 2-groups on measurable categories. Our paper pushes forward by studying representations of topological 2-groups, including Lie 2-groups. Topology really matters for infinite-dimensional representations. For example, it's a hopeless task to classify the infinite-dimensional unitary representations of even a little group like the circle, U(1). But it's easy to classify the continuous unitary representations.
A group has a category of representations, but a 2-group has a 2-category of representations! So, as usual, we have representations and maps between these , which physicists call "intertwining operators" or "intertwiners" for short. But we also have maps between intertwining operators, called "2-intertwiners".
This is what's really exciting about 2-group representation theory. Indeed, intertwiners between 2-group representations resemble group representations in many ways - a fact noticed by Elgueta. It turns out one can define direct sums and tensor products not only for 2-group representations, but also for intertwiners! One can also define "irreducibility" and "indecomposability", not just for representations, but also for intertwiners.
Our paper gives nice geometrical descriptions of these notions. Some of these can be seen as generalizing the following paper of Crane and Sheppeard:
17) Louis Crane and Marnie D. Sheppeard, 2-categorical Poincare representations and state sum applications, available as arXiv:math/0306440.
Crane and Sheppeard studied the 2-category of representations of the "Poincare 2-group". It turns out that we can get representations of the Poincare 2-group from discrete subgroups of the Lorentz group. Since the Lorentz group acts as symmetries of the hyperbolic plane, these subgroups come from symmetrical patterns like these:
18) Don Hatch, Hyperbolic planar tesselations, http://www.plunk.org/~hatch/HyperbolicTesselations/
But Crane and Sheppeard weren't just interested in beautiful geometry. They developed their example as part of an attempt to build new "spin foam models" in 4 dimensions. I've talked about such models on and off for many years here. The models I've discussed were usually based on representations of groups or quantum groups. Now we can build models using 2-groups, taking advantage of the fact that we have not just representations and intertwiners, but also 2-intertwiners. You can think of these models as discretized path integrals for gauge theories with a "gauge 2-group". To compute the path integral you take a 4-manifold, triangulate it, and label the edges by representations, the triangles by intertwiners, and the tetrahedra by 2-intertwiners. Then you compute a number for each 4-simplex, multiply all these numbers together, and sum the result over labellings.
Baratin and Freidel have done a lot of interesting computations in the Crane-Sheppeard model. I hope they publish their results sometime soon.
To wrap up, I'd like to make a few technical remarks about group representation theory and measurable fields of Hilbert spaces. In "week272" I talked about a class of measurable spaces called standard Borel spaces. Their definition was frighteningly general: any measurable space X whose measurable subsets are the Borel sets for some complete separable metric on X is called a "standard Borel space". But then I described a theorem saying these are all either countable or isomorphic to the real line! They are, in short, the "nice" measurable spaces - the ones we should content ourselves with studying.
In our work on 2-group representations, we always assume our measurable spaces are standard Borel spaces. We need this to get things done. But standard Borel spaces also show up ordinary group representation theory. Let me explain how!
To keep your eyes from glazing over, I'll write "rep" to mean a strongly continuous unitary representation of a topological group on a separable Hilbert space. And, I'll call an irreducible one of these guys an "irrep".
Mackey wanted to build all the reps of a topological group G starting from irreps. This will only work if G is nice. Since Haar measure is a crucial tool, he assumed G was locally compact and Hausdorff. Since he wanted L2(G) to be separable, he also assumed G was second countable.
For a group with all these properties - called an "lcsc group" by specialists wearing white lab coats and big horn-rimmed glasses - Mackey was able to construct a measure space G* called the "unitary dual" of G.
The idea is simple: the points of G* are isomorphism classes of irreps of G. But let's think about some special cases....
When G is a finite group, G* is a finite set.
When G is abelian group, not necessarily finite, G* is again an abelian group, called the "Pontryagin dual" of G. I talked about this a lot in "week273".
When G is both finite and abelian, so of course is G*.
But the tricky case is the general case, where G can be infinite and nonabelian! Here Mackey described a procedure which is a grand generalization of writing a rep as a direct sum of irreps.
If we choose a sigma-finite measure μ on G* and a measurable field Hx of Hilbert spaces on G*, we can build a rep of G. Here's how. Each point x of G* gives an irrep of G, say Rx. These form another measurable field of Hilbert spaces on G*. So, we can tensor Hx and Rx, and then form the "direct integral"
∫x (Hx ⊗ Rx) dμ(x)
As I already mentioned, a direct integral is a generalization of a direct sum. The result of doing this direct integral is a Hilbert space, and in this case it's a rep of G. The Hilbert spaces Hx specify the "multiplicity" of each irrep Rx in the representation we are building.
The big question is whether we get all the reps of G this way.
And the amazing answer, due to James Glimm, is: yes, if G* is a standard Borel space!
In this case we say G is "type I". People know lots of examples. For example, an lcsc group will be type I if it's compact, or abelian, or a connected real algebraic group, or a connected nilpotent Lie group. That covers a lot of ground. However, there are plenty of groups, even Lie groups, that aren't type I. The representation theory of these is more tricky!
If you want to know more, either read Mackey's book listed above, or this summary:
19) George W. Mackey, Infinite-dimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628-686. Available from Project Euclid at http://projecteuclid.org/euclid.bams/1183525453
The most fascinating thing about algebra and geometry is the way they struggle to help each other to emerge from the chaos of non-being, from those dark depths of subconscious where all roots of intellectual creativity reside. - Yuri Manin
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu
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