It’s been a while since I wrote the last entry, on representation theory of n-groups, partly because I’ve been polishing up a draft of a paper on a different subject. Now that I have it at a plateau where other people are looking at it, I’ll carry on with a more or less concrete description of the situation of a 2-group. For higher values of , describing things concretely would get very elaborate quite quickly, but interesting things already happen for . In particular, the case that I gave the talk about, a while back, was mostly the Poincaré 2-group, since this is the one Crane, Sheppeard, and Yetter talk about, and probably the one most interesting to physicists. It was first described by John Baez.
So what’s the Poincaré 2-group? To begin with, what’s a 2-group again?
I already said that a 2-group is a 2-category with only one object, and all morphisms and 2-morphisms invertible. That’s all very good for summing up the representation theory of as I described last time, but it’s sometimes more informative to describe the structure of concretely. A good tool for doing this is a crossed module. (A lot more on 2-groups can be found in Baez and Lauda’s HDA V, and there are some more references and information in this page by Ronald Brown, who’s done a lot to popularize crossed modules).
A crossed module has two layers, which correspond to the morphisms and 2-morphisms of . These can be represented as , where is the group of morphisms in , consists of the 2-morphisms ending at the identity of (a group under horizontal composition).
There has to be an action of on (morphisms can be composed “horizontally” with 2-morphisms), and a map (which picks out the source of the 2-morphism). The data have to fit together a certain way, which amounts to giving the axioms for a 2-category.
A handy way to remember the conditions is to realize that the action and the injection give ways for elements of to act on each other and for elements of to act on each other. These amount to doing first and then or vice versa, and both of these must amount to conjugation. That is:
Both of these are simplified in the case that maps everything in to the identity of – in this case, can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of . In this case, by the Eckmann-Hilton argument (the clearest explanation of which that I know being the one in TWF Week 100) it turns out that has to be commutative, so the first condition is trivial since , and the second is trivial since it follows from commutativity. This simpler situation is known as an automorphic 2-group.
In any case, given a 2-group represented as a crossed module, automorphic or not, the collection of all morphisms can be seen as a group in itself – namely the semidirect product , which is to say with the multiplication . “What?” you may ask, or maybe “Why?”
Maybe a concrete example would help, since we’d like one anyway: the Poincaré 2-group, which is an automorphic 2-group. There are versions of various signatures , in which case , and .
The group , then, consists of metric-preserving transformations of Minkowski space with the metric of signature – rotations and boosts (if any). The (abelian) group consists of translations of this space – in fact, being a vector space, it’s just a copy of it. Between them, they cover the basic types of transformation. Thinking of the translations as having a “projection” down to the identity rotation/boost may seem a bit artificial, except insofar as translations “don’t rotate” anything. More obvious is that rotations or boosts act on translations: the same translation can look differently in rotated/boosted coordinate systems – that is, to different observers.
So where does the Poincaré group come in? It’s the group of all metric-preserving transformations of Minkowski space, and is built from these two types: but how?
Well, the vector space is the group of transformations of the identity Lorentz transformation , since the map is trivial. But suppose that there is another copy of over each point in . Then we have the set of points , but notice that to talk about this as a group, we’d want a way to act on an element of one copy of over by another over . The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between and , which is that can act on , just as morphisms can act on 2-morphisms by “whiskering with the identity”. (Via Google books, here is the description of this in MacLane’s Categories for the Working Mathematician).
Concretely, this is the fact that there is a sensible way for both parts of to affect the , so we can say (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, , changes coordinates, and therefore the definition of the translation .
So that’s the construction of the Poincaré group from the Poincaré 2-group. What would be nice would be to have some clear description of some higher analog of Minkowski space where it makes sense to say the Poincaré 2-group acts as a 2-group. I don’t quite know how to set this up, but if anyone has thoughts, it would be interesting to hear them.
One reason is that, when describing representations of the 2-group, there’s an important role for spaces (or at least sets) with an action of the group – which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher -groups. Again – I don’t really know the answer to this. However, in Part 3 I’ll describe concretely how this works for 2-groups, and particularly the Poincaré 2-group.