### Synergism

#### Posted by David Corfield

When I first read in John’s writings about 2-groups I naïvely imagined that people would rapidly find analogues for everything done with ordinary groups. Hence my call for a Klein 2-geometry in 2006.

Later in that year, I’m to be found on the recently formed Café musing:

I presume people are wondering about which equivalents of features of group representation theory might be found? Are there ‘locally compact 2-groups’, and if so, are there Haar measure and Peter-Weyl theorem equivalents? For finite 2-groups, is there an equivalent of orthogonality in the character table? In general, is there an equivalent of the adjunction between the restricting and inducing functors? What about the branching rules? There must be dozens more questions like these.

To this John replied:

…right now there are very few people working on 2-groups. There are no “2-group theorists”: all these people are doing lots of other interesting things too. So, instead of trying to set up a vast edifice of machinery, the right approach is to find some really exciting examples, with connections to other branches of math, which will get more people interested in the subject. Then the machinery will practically build itself.

That’s why I’m working on specific examples, only developing enough general machinery to exhibit these examples. The string 2-group is the best one so far, since it hooks on to loop groups, affine Lie algebras, the WZW model - in short, the whole apparatus of postmodern Lie theory. The Poincare 2-group may also be really interesting - we’ll see.

Here I am, several months later, wondering if things couldn’t move along a little faster. Over in the other ‘culture’ it looks like they’ve just jointly solved a problem. Perhaps the ability to focus attention on a single problem was conducive to progress. On the other hand, having a couple of Fields’ medallists on the job must help.

Still, I’m wondering whether we’re maximising the synergy possible with the talented people who drop by here. John reported, again in 2006, that he and Jeff Morton

…figured out how to categorify the algebraic integers in any algebraic extension of the rationals, getting an “algebraic extension” of the category of finite sets. We figured out the beginnings of a theory that associates a “Galois 2-group” to any such algebraic extension.

I take it that work’s left idly gathering dust somewhere. Shouldn’t we be able to move that theory on quite rapidly?

Or, if there’s more interest in the representation theory of Lie 2-groups, then the observation that special functions, Bessel, hypergeometric, etc., appear as matrix entries of representations of particular Lie groups, might be suggestive. Does, say, the string 2-group have something special appear as matrix entries of representations?

## Re: Synergism

Well, I imagine that the representation theory of Lie 2-groups indeed does have interesting ‘matrix elements’… although I haven’t really thought of things in this way. After all, we know that the representation theory of the String 2-group is intimately tied up with the representation theory of loop groups. Instead of being a collection of numbers, the ‘matrix elements’ are now a collection of vector spaces — aren’t these related to the loop group modules graded by their energy? The identities between the special functions, like

now get replaced by

isomorphisms, according to the usual mantra. Aren’t these something like the fusion rules?The current strategy seems okay though: work out lots of interesting examples, and see what crops up.