## January 27, 2009

### Bruce Bartlett’s Thesis

#### Posted by John Baez

Bruce has left Sheffield and returned to South Africa… but his thesis has hit the arXiv, so if you get lonely for him, read this!

Abstract: This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended topological quantum field theory (TQFT), where the 2-category of unitary 2-representations of a finite group is thought of as the 2-category assigned to the point’ in the untwisted finite group model.

The first result is that the braided monoidal category of transformations of the identity on the 2-category of unitary 2-representations of a finite group computes as the category of conjugation equivariant vector bundles over the group equipped with the fusion tensor product. This result is consistent with the extended TQFT hypotheses of Baez and Dolan, since it establishes that the category assigned to the circle can be obtained as the higher trace of the identity’ of the 2-category assigned to the point.

The second result is about 2-characters of 2-representations, a concept which has been introduced independently by Ganter and Kapranov. It is shown that the 2-character of a unitary 2-representation can be made functorial with respect to morphisms of 2-representations, and that in fact the 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary equivariant vector bundles over the group.

The final result is about pivotal structures on fusion categories, with a view towards a conjecture made by Etingof, Nikshych and Ostrik. It is shown that a pivotal structure on a fusion category cannot exist unless certain involutions on the hom-sets are plus or minus the identity map, in which case a pivotal structure is the same thing as a twisted monoidal natural transformation of the identity functor on the category. Moreover the pivotal structure can be made spherical if and only if these signs can be removed.

Posted at January 27, 2009 4:42 PM UTC

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### Re: Bruce Bartlett’s Thesis

I’m glad you’ve got it on the Arxiv, Bruce! I’m sad to say I haven’t read more than the 2-Hilbert spaces chapter yet, despite the fact you gave me a sneak preview quite a while ago.

So, now that you’re an expert on it — what’s your hunch on the conjecture that all fusion categories can be made pivotal?

Posted by: Jamie Vicary on January 28, 2009 7:27 PM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

Thanks Jamie!

My problen when it comes to fusion categories is that all the nontrivial stuff can only occur in precisely that region which I know the least about — namely, when there are no inner products or $*$-structures sitting around. This was essentially proven by Etingof, Nikshych and Ostrik, but there’s also a simpler string-diagram-style proof in my thesis:

Prop 6.12, pg 127 (paraphrased). If a fusion category is equipped with a $*$-operation, then it can be made into a pivotal category.

Can you think of a fusion category (for readers: a fusion category is essentially just a semisimple linear monoidal category) which can’t carry a $*$-operation? Being more from physics and less algebraically inclined, my only intution for semisimple categories is for things like $\mathrm{Rep}\left(G\right)$ where $G$ is, say, a finite group. But since every representation of a finite group can be made unitary, we might as well define $\mathrm{Rep}\left(G\right)$ as the category of unitary representations of a group, in which case it certainly carries a $*$-structure (take the adjoint of a linear map).

The nontrivial fusion categories are things like the ‘Yang-Lee categories’ which ‘can be obtained as quotients of the categories of tilting modules over the quantum group ${U}_{q}\left({\mathrm{sl}}_{2}\right)$’ (see top of page 9 of Lectures on tensor categories) . My trouble is I don’t understand what a ‘tilting module’ is, I’m no good at this sort of representation theory, so I have no intuition about it.

By the way, I see that Brian Day released a paper on the arXiv today, A $*$-Autonomous Category of Banach Spaces, which is related to these kind of ideas.

Posted by: Bruce Bartlett on January 29, 2009 8:07 AM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

for readers: a fusion category is essentially just a semisimple linear monoidal category

What about the rigidity? That represents an enormous constraint on the monoidal structure.

If a fusion category is equipped with a $*$-operation, then it can be made into a pivotal category.

Very interesting, I didn’t know that.

Posted by: Jamie Vicary on January 29, 2009 10:28 AM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

What about the rigidity? That represents an enormous constraint on the monoidal structure.

Woops sorry, you’re right. Ok, let’s try: A fusion category is a semisimple linear rigid monoidal category with only finitely many isomorphism classes of simple objects, and where the unit object $1$ is simple.

Posted by: Bruce Bartlett on January 29, 2009 11:59 AM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

This is a question about fusion categories and representation categories of quantum groups. I know Bruce doesn’t talk about quantum groups much in his thesis, but at least I know he’s interested in them, so I’m sure he won’t mind!

I want to know the current status of the conjecture that every braided fusion category over the complex numbers has a braided monoidal equivalence to the category of representations of a quantum group, or perhaps of a quantum group with certain straightforward properties.

Bruce gave a definition of a fusion category above, but just to have everything in one place, here’s another one: a fusion category over the complex numbers is a $ℂ$-linear semisimple monoidal category with duals, with finitely many simple objects and finite-dimensional hom-sets, such that the endomorphism algebra of the tensor unit is $ℂ$. It’s possible that we’ll actually need Hilb—fusion categories, ones for which the hom-sets are actually Hilbert spaces, and for each morphism in the category we can find its adjoint in a compatible way. We would then be dealing with braided monoidal 2-Hilbert spaces with duals.

I have long wondered why nobody talks about this conjecture very much! This seems like an obvious sort of representation theorem to think about, since in the symmetric case these categories are known to be the categories of representations of finite supergroups, by the Doplicher-Roberts theorem. (A supergroup is just a group with a chosen element that commutes with everything.)

I can’t find much about this conjecture on the internet, which I think is pretty weird. A representation theorem has been described by Hendryk Pfeiffer for modular categories, which seems to be the sort of thing that I’m looking for, except that it only applies to fusion categories that are modular — a condition that can loosely be described as “definitely not symmetric”. I find this a bit weird, given that it’s precisely in the symmetric case that this sort of representation theorem was first established by Doplicher and Roberts! I’m not sure whether there are any fusion categories of the sort that I’m interested in which are neither symmetric and nor modular.

So, does anyone know what the feeling is about this among those in the know? Maybe it’s a big secret that nobody’s blogging about. Or maybe nobody cares! Anyway, I’d like to know what’s going on here.

Posted by: Jamie Vicary on February 17, 2009 6:44 PM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

Jamie wrote:

I want to know the current status of the conjecture that every braided fusion category over the complex numbers has a braided monoidal equivalence to the category of representations of a quantum group, or perhaps of a quantum group with certain straightforward properties.

I don’t know the status of this. I would like to know!

What do you mean by ‘quantum group’? It’s pretty common these days to limit ‘quantum group’ to mean ‘$q$-deformed universal enveloping algebra of a semisimple Lie algebra’, but you must mean something much broader, more like ‘quasitriangular Hopf algebra’. Do you have a specific conjecture in mind, or are you trying to keep your options open?

Jamie wrote:

I have long wondered why nobody talks about this conjecture very much!

I thought it was because it was hard to build a good forgetful functor from your braided fusion category to $\mathrm{Vect}$.

But people have talked about this problem, and I don’t really know what they’ve said. There’s Hendryk Pfeiffer’s paper, which has some references to others…

Jamie wrote:

I’m not sure whether there are any fusion categories of the sort that I’m interested in which are neither symmetric and nor modular.

I think there must be, because somewhere Michael Müger has described a process for forcing a certain class of ‘pre-modular categories’ to become modular, and if you do this forcing process ‘part-way’, or not at all, you should get something non-modular but not symmetric either.

I also recommend that you scour this very nice paper for interesting facts:

It starts out easy but towards the end it gets into some fun stuff.

Posted by: John Baez on February 17, 2009 9:05 PM | Permalink | Reply to this

### Re: Bruce Bartlett’s Thesis

Thanks for that reply, John. By ‘quantum group’ I mean some sort of quasitriangular Hopf algebra. I certainly don’t have a conjecture in mind! Thanks for suggesting that link, I’ll check it out.

As you say, I suppose the difficulty comes down to the fact that it’s not easy to construct a canonical forgetful functor to Vect. So perhaps this gives us a reason to doubt the conjecture. But more importantly, as far as I can tell, nobody has found a counterexample! Googling around, there are a few people who seem to talk about counterexamples, but they are always working in a slightly different context (for example, working over some other field rather than over the complex numbers.) This can dramatically change things — for example, the theory of real representations of finite groups is quite different to the theory of their complex representations.

Physically, it seems to me that the categories of the sort I described are by far the most natural ones to consider. For example, just today I was talking to Rick Blute who’s visiting Oxford for a few months, and he was explaining to me how these categories arise as the categories of ‘vacuum perturbations’ in algebraic quantum field theory on Minkowski space! In a spacetime dimension of 4 or more, this category is symmetric monoidal, but in a spacetime dimension of 3 it’s only braided. So in a sense, classifying these categories is the same as classifying 3-dimensional quantum field theories.

There are only two possibilities: either I’m mistaken about the importance of these braided monoidal 2-H*-algebras; or the mathematicians that think about these things aren’t aware of their importance, and aren’t giving them the attention they deserve. Either way, more discussion is needed.

Posted by: Jamie Vicary on February 23, 2009 6:12 PM | Permalink | Reply to this
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