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 n, describing things concretely would get very elaborate quite quickly, but interesting things already happen for n=2. 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 \mathbb{G} 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 \mathbb{G} as I described last time, but it’s sometimes more informative to describe the structure of \mathbb{G} 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 \mathbb{G}. These can be represented as (G,H,\triangleright, \partial), where G is the group of morphisms in \mathbb{G}, H consists of the 2-morphisms ending at the identity of G (a group under horizontal composition).

There has to be an action \triangleright : G \rightarrow End(H) of G on H (morphisms can be composed “horizontally” with 2-morphisms), and a map \partial : H \rightarrow G (which picks out the source of the 2-morphism). The data (G,H,\triangleright,\partial) 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 \triangleright : G \rightarrow End(H) and the injection \partial : H \rightarrow G give ways for elements of G to act on each other and for elements of H to act on each other. These amount to doing first \triangleright and then \partial or vice versa, and both of these must amount to conjugation. That is:

\partial(g \triangleright h) = g (\partial h) g^{-1}

and

(\partial h_1) \triangleright h_2 = h_1 h_2 h_2^{-1}

Both of these are simplified in the case that \partial maps everything in H to the identity of G – in this case, H can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of \mathbb{G}. 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 H has to be commutative, so the first condition is trivial since \partial h = 1, 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 G \ltimes H, which is to say G \times H with the multiplication (g_1,h_1) \cdot (g_2,h_2) = (g_1 g_2 , g_2 \triangleright h_1 h_2). “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 (p,q), in which case G = SO(p,q), and H = \mathbb{R}^{p+q}.

The group G, then, consists of metric-preserving transformations of Minkowski space R^{p+q} with the metric of signature (p,q) – rotations and boosts (if any). The (abelian) group H 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 SO(p,q) \ltimes \mathbb{R}^{p+q} 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 H = \mathbb{R}^{p+q} is the group of transformations of the identity Lorentz transformation 1 \in G = SO(p,q), since the map \partial : H \rightarrow G is trivial. But suppose that there is another copy of H over each point in G. Then we have the set of points G \times H, but notice that to talk about this as a group, we’d want a way to act on an element h_1 of one copy of H over g_1 \in G by another h_2 over g_2. The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between G and H, which is that G can act on H, 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 (g_1,h_1) to affect the h_2, so we can say (g_2,h_2) \cdot (g_1,h_1) = (g_2 g_1, g_1 h_2 + h_1) (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, g_1, changes coordinates, and therefore the definition of the translation h_2.

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 G – which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher n-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.

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