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December 24, 2008

Infinite-Dimensional Representations of 2-Groups

Posted by John Baez

Yay! This paper is almost ready for the arXiv! We’ve been working on it for years… it turned out to involve a lot more measure theory than we first imagined it would:

Here’s what it’s about:

A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called ‘measurable categories’ (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners — features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify ‘irretractable’ representations — another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered ‘separable 2-Hilbert spaces’, and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

I’ll try to explain this stuff soon in This Week’s Finds. Right now I’ll just throw it out there as a Christmas present to the universe. If you catch typos or other errors, please let me know.

It’s long, but that’s because it’s self-contained. We start by explaining what a 2-category is, what a 2-group is, what a Kapranov–Voevodsky 2-vector space is, what a measurable category is, and so on… and there are big fat appendices explaining all the measure theory. So, anyone who has taken a class in real analysis and a class in group theory can read this — if trapped with nothing else to do for a sufficiently long time.

Speaking of which: the weather in most of the USA has been dreadful over the last few weeks: blizzard, ice storms, even clouds here in Riverside. Lots of flights have been delayed or canceled; lots of roads have been shut down. So, if you’ve had trouble traveling, you have my sympathies. And if you’re about to fly somewhere, print out a copy of this paper and take it with you! By the time you reach your destination, you may be an expert on 2-group representation theory.

Merry Christmas! Happy Hanukkah! Cool Kwanzaa! Wild Winter Solstice! And a fantastic Festivus!

Posted at December 24, 2008 7:25 PM UTC

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12 Comments & 1 Trackback

Re: Infinite-Dimensional Representations of 2-Groups

Great! Santa’s been and it’s not even midnight. I always thought it was just John dressed up.

Happy Christmas, everyone.

Posted by: David Corfield on December 24, 2008 10:35 PM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

And a great Agnostica!

Posted by: Toby Bartels on December 25, 2008 2:00 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

You blew my cover, David! It’s true: I’m Santa. I usually call myself the Wizard, but if anyone who paid close attention would have noticed the suspicious similarity.

Ho ho ho!

Posted by: John Baez on December 25, 2008 7:46 AM | Permalink | Reply to this

Santa Claus LLC; Re: Infinite-Dimensional Representations of 2-Groups

I managed to quote some of Yuri I. Manin’s Scheme-based advocacy that we should take the terminology “arithmetic surface” for spec(Z[x]) a lot more seriously, in:

Santa Claus LLC, Minutes of Executive Committee, 24 December 2008

which is my Xmas card to you and yours.

Posted by: Jonathan Vos Post on December 25, 2008 6:00 PM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

If things went as well as you hoped with this work and it found physical application, would there be a parallel to much of what Wigner tells us in On Unitary Representations of the Inhomogeneous Lorentz Group? For example,

We see thus that there corresponds to every invariant quantum mechanical system of equations such a representation of the inhomogeneous Lorentz group. This representation, on the other hand, though not sufficient to replace the quantum mechanical equations entirely, can replace them to a large extent… the representation can replace the equation of motion, it cannot replace, however, connections holding between operators at one instant of time.

Posted by: David Corfield on February 6, 2009 9:26 AM | Permalink | Reply to this
Read the post Where is the Philosophy of Physics?
Weblog: The n-Category Café
Excerpt: Should we devote more time to the philosophy of physics?
Tracked: May 19, 2009 2:17 PM

Re: Infinite-Dimensional Representations of 2-Groups

More representation theory of 2-groups today – On the regular representation of an (essentially) finite 2-group – from Josep Elgueta.

The regular representation of an essentially finite 2-group 𝔾\mathbb{G} in the 2-category 2Vect k\mathbf{2Vect}_k of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in Rep 2Vect k(𝔾)\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G}) are 2-vector spaces under quite standard assumptions on the field kk, and a formula giving the corresponding “intertwining numbers” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ω:Rep 2Vect k(𝔾)2Vect k\mathbf{\omega}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G}) \to \mathbf{2Vect}_k is representable with the regular representation as representing object. As a consequence we obtain a kk-linear equivalence between the 2-vector space Vect k 𝒢\mathbf{Vect}_k^{\mathcal{G}} of functors from the underlying groupoid of 𝔾\mathbb{G} to Vect k\mathbf{Vect}_k, on the one hand, and the kk-linear category nd(ω)\mathcal{E} n d (\mathbf{\omega}) of pseudonatural endomorphisms of ω\mathbf{\omega}, on the other hand. We conclude that nd(ω)\mathcal{E} n d (\mathbf{\omega}) is a 2-vector space, and we (partially) describe a basis of it.

Posted by: David Corfield on July 7, 2009 11:49 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

Infinite-Dimensional Representations of 2-Groups was finally accepted for publication as a book in the series Memoirs of the AMS. Science proceeds apace.

Posted by: John Baez on May 27, 2011 5:55 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

Congratulations! I would like to persuade my library to buy one.

Posted by: David Roberts on May 27, 2011 8:28 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

Thanks! You have lots of time, because it was only accepted for publication today.

One good thing about publishing a book in Memoirs of the American Mathematical Society is that many American universities automatically buy all these books. But I guess not Australian universities.

By the way, Varghese Mathai has invited me to Adelaide. If I came there sometime between September and December this year, would you be there then?

Posted by: John Baez on May 27, 2011 9:19 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

This book has finally been published by the AMS, and you can get a nice PDF file that looks like the book here.

I mention this in part because I’m relieved the book is finally done, but also because it may be a limited-time offer. I never heard that the AMS gives away free e-book versions of their Memoirs, so their doing so may be a mistake that they’ll fix later. I don’t know, but enjoy it while you can.

The arXiv version is, of course, free forever.

Posted by: John Baez on June 25, 2012 5:55 PM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

Hmm, not free to me at the moment.

(Oh, and apologies for not replying to your message of May 27 last year. I was in town, but I didn’t hear of you coming, else I would have caught up with you.)

Posted by: David Roberts on June 26, 2012 12:25 AM | Permalink | Reply to this

Re: Infinite-Dimensional Representations of 2-Groups

… but it is if I use Adelaide’s library proxy. There you go.

Posted by: David Roberts on June 26, 2012 12:35 AM | Permalink | Reply to this

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