The n-Category Café

I paraphrased one of the main ideas of the toposophers of quantum mechanics by saying:

…given any noncommutative structure A, we may try to “cover” it by abelian things.

David Corfield asks:

Is this perspective then at odds with Connes’, who seems to enjoy noncommutative structures for what they are?

At least the kind of “abelian cover” that I mentioned is a rather different concept compared to the abelian patching that Alain Connes is referring to in the paragraph that you quote.

It might be useful to think of it this way:

the nature of the “abelian cover” of $A$ that I mentioned, namely the precise nature of the category $C(A)$, is actually invisible. Or supposed to be. What we really want to say is that

The theory of noncommutative algebras in a topos $S$ is equivalent to the theory of commutative algebras in some topos $T$ internal to $S$.

That the topos $T$ in question can be modeled by considering presheaves on a category of abelian subalgebras of $A$ might be useful for explicit computations, but is actually not something that pertains to the intrinsic meaning of the above statement, I would say.

So the way I am thinking of it, I would actually regard the toposopher’s perspective as a strengthening of Alain Connes’ point of view:

they kind of say: yes, Connes is precisely right in that the noncommutative algebra $A$ doesn’t have to be thought of as being patched together from abelian things. And we can actually make this precise: because the noncommutative algebra is one single commutative algebra – when regarded in the right context.

Why are we so eager to abelianize everything?

Let’s qualify this question a little: why are the toposophers of quantum mechanics so eager to abelianize everything?

(Because in other areas people are equally eager to make everthing in sight non-abelian ;-)

And the answer is: because they think this is the right way to deal with the vexing measurement problem of quantum mechanics.

The non-abelianness of the quantum phase space implies that the connection between the corresponding physical model and its interpretation is subtle. The toposophers say:

this subtlety is precisely the need to interpret everything internal to a suitable topos. And the construction of this happens to involve, if you look at it externally, “local abelian covers” of the nonabelian situation.

Okay, that’s what I can say. Now I’ll think about how to say it better. And probably before I can do that John Baez will chime in and give a crystal clear account.

David wrote:

Why are we so eager to abelianize everything?

I think it’s because people like situations where a state of a system simply consists of a state for each part. They like situations where we can freely duplicate and delete information. In other words, they like symmetric monoidal categories that are cartesian. The classic examples are the category of sets and various categories of ‘spaces’. There’s something very geometrical about cartesian categories.

Algebras form a symmetric monoidal category with their usual tensor product, but it’s not cartesian.

Commutative algebras also form a symmetric monoidal category with their usual tensor product, and this isn’t cartesian either. But, the opposite of this category is cartesian!

In other words, commutative algebras are dual to ‘spaces’ — or technically, ‘affine schemes’ — and affine schemes form a cartesian category. This is the idea behind algebraic geometry, which secretly goes back to Descartes.

Noncommutative algebras are just different. I’ve always thought we should just accept them for what they are. But, it’s interesting that we can often think of them as giving rise to commutative algebras not inside the category of sets, but inside some other cartesian category!

You can think of this trick as a clever way to keep ahold of the cartesianness we love so much.

Maarten wrote:

I think the oldest example of the idea of studying something non Abelian by looking at Abelian subthingies might be the theory of simple Lie algebras, where you use the Cartan subalgebra […] to analyse the structure and representations. In this case all Abelian subthingies are conjugate, so probably there is no need for topoi.

But it’s still possible that topoi could help clarify what’s really going on, at least for people who understand topoi.

When, shockingly late in life, I first seriously tried to understand simple Lie algebras and their representations, I spent some time trying to figure out the deep inner meaning of the usual Cartan subalgebra / Weyl group / weight lattice baloney.

I call it “baloney” because while it’s great, everyone seems to blindly accept it, without asking why it really works. No matter how good a piece of math is, we need to keep rethinking it, or it becomes stale.

I didn’t get too far, but I had hopes. I was thinking about Lie groups rather than Lie algebras, but that shouldn’t matter much. I was trying to see a compact simple Lie group $G$ as somehow ‘built from’ the action of the Weyl group $W$ on a maximal torus $T$. And, I wanted to use this to describe the usual category $Rep(G)$ of finite-dimensional representations of $G$ in terms of the action of $W$ on $Rep(T)$. $Rep(T)$ is equivalent to the category of vector bundles on the dual group $T^*$, also known as the “weight lattice”. So, I hoped this stuff would give a nice conceptual approach to the usual “highest weight” approach to constructing irreducible guys in $Rep(G)$.

I didn’t make much progress, so I posted a question about it to sci.math.research:

I’m enjoying Adams’ book on Lie groups, where he downplays the semisimple Lie theory and emphasizes compact Lie groups in comparison to standard treatments. One of the basic results is as follows.

Suppose $G$ is a compact Lie group, $T$ a maximal torus, and $W$ the Weyl group (the normalizer of $T$ mod the centralizer of $T$). Then the following two algebras are isomorphic: the algebra of class functions on $G$, and the algebra of $W$-invariant class functions on $T$, where W acts as automorphisms of $T$ in the obvious way. The isomorphism is given by restricting class functions on $G$ to class functions on $T$.

Now the algebra of class functions on $G$ is also called the representation ring of $G$, for good reasons. The category of representations of $G$ is a monoidal category with a notion of direct sums, and from any such category one can extract an algebra whose elements are formal linear combinations of objects; direct sums in the category correspond to addition in the algebra, and tensor products in the category to products in the algebra, in a well-known way. (This is the Grothendieck ring construction.) Each representation of G yields an element of the representation ring, and concretely speaking, the latter is just the character of the representation.

So I suspect the following. Let $C$ be the category of finite-dimensional representations of $G$. Then there is some category $C^'$ of representations of $T$ equipped with some extra bit of structure involving $W$, such that $C$ and $C^'$ are equivalent as monoidal categories with direct sums. (To be more formal, instead of “with direct sums” I could talk about abelian categories, but it’s not those category-theoretic niceties that are the issue here, it’s the group theory.)

To get from $C$ to $C^'$, we first of all simply restrict any representation of $G$ to a representation of $T$. But then we need to retain some extra bit of structure involving $W$ to make sure we don’t lose any information (so that $C$ and $C^'$ are equivalent). What is it? Whatever it is, it’s reflected in the fact that when we take the character of a representation of $G$ and restrict it to $T$, we get something $W$-invariant.

The process of getting back from $C^'$ to $C$ is closely related in spirit to the “highest weight representation” construction. But I’d like to talk about it in a way that doesn’t use any arbitrary choices (like a choice of Weyl chamber).

I got a reply from Gavin Wraith which suggested thinking of $T$ as a group in the topos of $W$-sets, and trying to use this as a kind of stand-in for $G$. It’s easy to see that $Rep(T)$ becomes a category in the topos of $W$-sets. But, I don’t think we got as far as we wanted, namely to describe $Rep(G)$ in this language.

However, what’s nice is that we see a way to get from a compact connected Lie group to a compact connected abelian Lie group in a certain topos.

This is almost exactly analogous to what Heunen and Spitters do for C*-algebras!

The main difference is that Wraith considers a topos of presheaves on a certain groupoid, while Heunen and Spitters consider a topos of presheaves on a certain poset. Wraith’s groupoid consists of maximal abelian subgroups of the given Lie group, with ‘conjugations’ as isomorphisms. Heunen and Spitters’ poset consists of all abelian C*-subalgebras of the given C*-algebra, with inclusions as morphisms.

Maybe we should try to get the benefits of both approaches, by working with presheaves on a category where the morphisms are ‘inner monomorphisms’ — that is, conjugations followed by inclusions.

I think also for Kac-Moody algebras Cartans are not always conjugate

KM algebras have at least unique Dynkin diagrams, so Cartan algebras are isomorphic. In contrast, most infinite-dimensional Lie algebras, e.g. Virasoro, don’t have Dynkin diagrams and Cartan matrices; twice a root can still be a root. For Lie superalgebras the situation is even more complex even in the finite-dimensional case: some have a unique Dynkin, some have several, and some have none, e.g. diff algebras in (0|m) dimensions.