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.