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May 4, 2012

Quivering with Excitement

Posted by John Baez

Over on Google+, David Roberts just told me the most exciting theorem I’ve heard all week. Every projective variety is the Grassmannian of a quiver representation! I suppose it’s just another indication of the ‘wildness’ of quiver representations once we leave the safe waters of Gabriel’s theorem.

Let me explain….

To briefly recall: a quiver is a category QQ freely generated by a finite directed graph. A quiver representation is a functor F:Q→FinVectF: Q \to FinVect. In other words, it’s just a finite-dimensional vector space for each vertex and a linear operator for each edge. A morphism of quiver representations is a natural transformation between functors of this sort. Again, this amounts to something pathetically simple.

The first surprise:

Gabriel’s Theorem. A quiver has finitely many isomorphism classes of indecomposable representations iff its underlying undirected graph is a Dynkin diagram of type A, D, or E.

It’s fun to work out examples of this theorem and delve deeper into the relation between quivers and Lie theory. It’s the tip of an iceberg that mathematicians plan to mine until they hit bottom and sea water rushes in and drowns them all.

But now:

Given a quiver QQ and representations F,GF, G, define the Grassmanian Gr(Q,F,G)Gr(Q,F,G) to be the set of monomorphisms from GG to FF, modulo the action of automorphisms of G. (We can compose a mono f:G→Ff: G \to F with an automorphism g:G→Gg: G \to G to get a new mono fg:G→Ff g: G \to F.)

To convince yourself that this deserves to be called a Grassmannian, look at the case where QQ comes from the graph with a single vertex and no edges. Then a representation of QQ is a vector space, a morphism of representations is a linear map, and a Grassmannian consists of all nn-dimensional subspaces of an mm-dimensional vector space.

Lemma. For any quiver, the Grassmannian Gr(Q,F,G)Gr(Q,F,G) can be made into a projective variety in some systematic way.

Reineke’s Theorem. Every projective variety arises this way for some quiver with at most 3 vertices.

The title of Reineke’s paper is Every projective variety is a quiver Grassmannian. The abstract says merely “The theorem is proved.” The paper is 2 pages long. This is a man who doesn’t beat around the bush.

For more, try Lieven Lebruyn’s blog article Quiver representations can be anything.

Despite the exciting nature of this result, I have no idea what it might be ‘good for’. Is it fun or helpful to classify projective varieties in terms of the quivers that give them?

Posted at May 4, 2012 6:20 AM UTC

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Re: Quivering with Excitement

Perhaps something that might be interesting is relating the stratification of the variety by its singularities to the structure of the quiver. A cryptic comment by (I presume) Thomas Riepe at Lieven’s blog post links to this MO question: Grothendieck’s manuscript on topology. One thing that Grothendieck was interested in in the mid 1980s was finding a new basis for topology that was much more geometrically oriented than point-set topology (he called it ‘tame’ topology, I believe), and for which archetypal examples would be varieties and moduli spaces with their natural stratifications. I wonder if using generalisations of quiver Grassmannians leads to other interesting ‘spaces’.

Another interesting thing would be to relate cohomological invariants of varieties (basically, motives) to those of quivers.

There are a whole stack of interesting papers by Reineke dealing with quivers.

Posted by: David Roberts on May 4, 2012 7:03 AM | Permalink | Reply to this

Re: Quivering with Excitement

Well actually the first question is ask is how functorial this whole process is. Given a map of quiver reps we must get a map of varieties. But what is this functor like? Does it have an adjoint? It depends on what sort of category of varieties one considers. If we take seriously the idea that cohomological invariants of quiver reps and varieties are related, then it makes sense to consider the sorts of morphisms between varieties that motivic people do (correspondences of finite type or something like that).

Posted by: David Roberts on May 4, 2012 7:10 AM | Permalink | Reply to this

Re: Quivering with Excitement

Quibbling, isn’t the term quiver used for the directed graph rather than the category it generates?

More interestingly, perhaps, are there any distinguishing, nice features of the representation theory of quivers with underlying Dynkin diagrams which are not A, D or E? E.g., for B, are things only slightly worse than for A, D, E?

Posted by: David Corfield on May 4, 2012 10:46 AM | Permalink | Reply to this

Re: Quivering with Excitement

David wrote:

Quibbling, isn’t the term quiver used for the directed graph rather than the category it generates?

Yes, that’s what everyone else seems to say, and if you roll back that nLab page long enough you’ll find that someone else changed my original definition of quiver to bring it in line with the norm. But I find it sad to waste a nice word like ‘quiver’ to mean exactly the same thing as ‘directed graph’. So, I use it to mean a category QQ freely generated by a directed graph. As an added bonus, I then get to say a representation of the quiver QQ is just a functor F:Q→FinVectF : Q \to FinVect. Just a matter of taste, no big deal.

More interestingly, perhaps, are there any distinguishing, nice features of the representation theory of quivers with underlying Dynkin diagrams which are not A,DA, D or EE?

Yes, but the main result I know goes in a slightly different direction than you guessed.

E.g., for BB, are things only slightly worse than for A,D,EA, D, E?

Actually, the BB Dynkin diagram is not really a graph: what looks like a ‘double edge’ in that diagram is not really two edges, just an abbreviation for an edge with some extra data attached to it, namely the number ‘4’.

(Triple edges stand for the number 6, double edges stand for the number 4, single edges stand for the number 3, and invisible edges stand for the number 2. Don’t blame me.)

Next best after the quivers coming from ADE Dynkin diagrams are those coming from affine ADE Dynkin diagrams:

namely the ones called A˜,D˜\tilde{A}, \tilde{D} and E˜\tilde{E} here. These affine Dynkin diagrams are related to centrally extended loop groups of simple Lie groups much as the ordinary kind are related to simple Lie groups.

And if I’m remembering correctly, there’s an addendum to Gabriel’s theorem saying that the A˜,D˜\tilde{A}, \tilde{D} and E˜\tilde{E} quivers are precisely the ones that have, not finitely many indecomposable representations, but the next best thing: a ‘tame’ collection of indecomposable representations. This means that those representations are parametrized by a nice finite-dimensional space: an algebraic variety, in fact.

The rest of the quivers are said to be ‘wild’: roughly, their collection of indecomposable representations is some sort of intimidating infinite-dimensional space.

There is also a theory that deals with the Dynkin diagrams of types B,C,FB, C, F and GG, but I’ve never quite understood it. It involves quivers with dots labelled by different division algebra, or something like that.

Posted by: John Baez on May 4, 2012 11:27 AM | Permalink | Reply to this

Re: Quivering with Excitement

This ‘tame versus wild’ dichotomy is something I imagine you might like, David:

The notions of tame and wild problems is now rather popular in various branches of representation theory and related topics, especially because of the so-called tame-wild dichotomy (cf. e.g. [3, 7] and other papers). Namely, in most cases it so happens that either indecomposable representations depend on at most one parameter or their description becomes in some sense “universal,” i.e. containing a classification of representations of all finitely generated algebras.

This is from the paper Tame-wild dichotomy for derived categories by Viktor I. Bekkert, Yuriy A. Drozd.

So maybe when I said the ‘tame’ quivers had indecomposable representations parametrized by an algebraic variety I was understating quite how tame they are! On the other hand, classifying representations of all finitely generated algebras is worse than a Herculean task: more like Sisyphean.

Posted by: John Baez on May 4, 2012 11:40 AM | Permalink | Reply to this

Re: Quivering with Excitement

the BB Dynkin diagram is not really a graph

Whoops, that’s a worrying erosion of memory. I used to know a little about angles in root spaces.

It involves quivers with dots labelled by different division algebra, or something like that.

Sounds like it would have been your scene, once. Valued quivers seem to have something to do with it.

Posted by: David Corfield on May 4, 2012 12:13 PM | Permalink | Reply to this

Re: Quivering with Excitement

I think the correct buzzword is modulated quiver. In his book Representations and Cohomology I, D. J. Benson writes

A modulated quiver consists of a valued graph together with an orientation and a modulation.

where a modulation is an assignment of a division ring to each vertex and a bimodule to each edge. I’ve never understood how this lets you generalize stuff from the simply-laced (that is, ADE) case to the general case, and I’ve never understood exactly how much stuff it lets you generalize.

And I probably never will. With pure math there’s always a taller mountain lurking behind the one you just climbed, beckoning yet forbidding. So it’s a very interesting feeling to shift from the taking the attitude “someday I’ll climb that one too” to taking the attitude “nah, I’ve had enough”. For many mathematicians I suppose this shift happens when they get old and tired. But to quit because you want to try something else is a bit different. You have to decide how much you’re willing to simply forget, and how much you’re willing to keep pursuing in a more hobby-esque way.

Posted by: John Baez on May 5, 2012 9:01 AM | Permalink | Reply to this

Re: Quivering with Excitement

One way that I like to think about quivers and valued quivers is via tensor algebras. Take a semisimple algebra A and an A-bimodule M, and form the tensor algebra T_A(M), so the graded algebra having the n-fold tensor product of M over A in degree n.

If A is just a product of copies of a field, then this is the path algebra of a quiver, and conversely. More generally, you can take A to be a product of division rings, and then you basically have the path algebra of a valued quiver. As a simple example, you can get type B_2 by taking M=C (complex numbers), viewed as an R-C-bimodule, and then ‘restricting scalars’ to view M as a bimodule for A=RxC. The tensor algebra is nothing more than upper-triangular 2×2 matrices over C, but with the top left entry real.

From this perspective, it is clear that (pretty much) everything you know about quiver representations goes through to this more general case (at least, assuming A and M are both finite dimensional over some fixed base field). It also has the advantage of doing away with the surplus of indices that crop up when thinking about valued quivers, modulations, etc.

Posted by: andrew hubery on May 9, 2012 9:30 PM | Permalink | Reply to this

Re: Quivering with Excitement

I find it sad to waste a nice word like ‘quiver’ to mean exactly the same thing as ‘directed graph’

There was some extensive discussion of this on the nForum before the change was made. I believe the conclusion was that (1) it’s better not to alienate quiver-theorists and (2) graph theorists often use ‘directed graph’ in a different way to category-theorists, so it’s nevertheless useful to have a word which unambiguously means the category-theorists’ version (namely, one which allows, but does not mandate, arbitrary parallel edges and loops).

Posted by: Mike Shulman on May 4, 2012 7:40 PM | Permalink | Reply to this

Terminology

(1) it’s better not to alienate quiver-theorists and (2) graph theorists often use ‘directed graph’ in a different way

One could add: (3) saying “quiver” nicely indicates the context in which the directed graph will be used.

Compare to “presheaf”. One can argue in the same way that it is a waste to use this term to mean exactly the same thing as ‘contravariant functor’. But it still has its good use: it indicates that one is going to think of contravariant functors in a very specific way.

Posted by: Urs Schreiber on May 4, 2012 9:28 PM | Permalink | Reply to this

Re: Quivering with Excitement

Mike wrote:

… it’s nevertheless useful to have a word which unambiguously means the category-theorists’ version (namely, one which allows, but does not mandate, arbitrary parallel edges and loops).

Since there’s an annoyingly large multiplicity of meanings of the word ‘graph’, it would be great if all category theorists rallied around using ‘quiver’ to mean ‘directed graph that allows, but does not mandate, parallel edges and loops’. But alas, it’s not going to happen: all the category theorists I know speak of the ‘underlying graph of a category’, not the ‘underlying quiver of a category’.

But I’m not arguing against your decision! I’m just glad I’m not part of any sort of committee-like body that decides on optimal math terminology: I prefer to focus on pleasing myself.

Posted by: John Baez on May 5, 2012 8:30 AM | Permalink | Reply to this

Re: Quivering with Excitement

Obviously, it’s a cool theorem, but I think you may be blowing it out of proportion a bit; this seems to be one more manifestation of Murphy’s law in algebraic geometry. It’s very interesting observation that any class of moduli spaces will either very well-behaved or literally anything possible, but this is just one more piece of evidence for this phenomenon.

Posted by: Ben Webster on May 4, 2012 2:53 PM | Permalink | Reply to this

Re: Quivering with Excitement

I guess I’d call this the tame/wild dichotomy. Longfellow wrote about it:

And when she was good,
She was very, very good
But when she was bad she was horrid.

Posted by: John Baez on May 5, 2012 9:34 AM | Permalink | Reply to this

Re: Quivering with Excitement

this seems to be one more manifestation of Murphy’s law in algebraic geometry

I thought John’s perspective was different from that:

Of course when you come from the point of view of looking at moduli spaces of quiver Grassmannian’s, then the result tells you that it’s disappointing: nothing specific can be said about these moduli spaces.

But I thought what John found interesting is that you might turn this around and say: hey, the naive idea that you might have when you first learn about projective varieties, namely that they are all like Grassmannian’s, actually is true, if only you allow quiver Grassmannians.

I envisioned him next speculating about reformulating the foundations of algebraic geometry once again, this time as “geometry modeled on quiver Grassmannians”. And that this is what he found exciting. (I might be wrong as he hasn’t told us what he finds exciting about the result, it seems.)

But of course I don’t know. I just thought I’d point out that you might be talking past each other.

Posted by: Urs Schreiber on May 5, 2012 2:18 PM | Permalink | Reply to this

Re: Quivering with Excitement

I just found it surprising at first that quiver Grassmannians, which sound so special at first, are so general. Of course this raises the hope that one can do something with this fact, like understand something new about projective varieties. But I have no idea how, so this is only the most exciting theorem I’ve heard all week—not all month. Indeed, I heard a better one last week.

Posted by: John Baez on May 6, 2012 6:02 AM | Permalink | Reply to this

Re: Quivering with Excitement

This reminds me of a comment of Peter Johnstone’s about the Freyd–Mitchell embedding theorem (every abelian category is a nice subcategory of RR-ModMod, for some ring RR).

Often the theorem is used in the following way: we have an abstract abelian category, and the Freyd–Mitchell theorem lets us assume that it’s actually a category of modules. This gives us extra concreteness.

But Peter prefers to read it the other way round: given a category of modules, you might as well forget that it’s a category of modules, remembering only that it’s an abelian category. This removes distracting extra structure and forces you to focus on the essential.

Similarly, you could take advantage of this result of Reineke in two ways: if you’re trying to prove something about an arbitrary projective variety XX, you could add concreteness by using the fact that XX is a quiver Grassmannian. Alternatively, if you’re trying to prove something about quiver Grassmannians, you could clear away conceptual clutter by working instead with abstract projective varieties.

Posted by: Tom Leinster on May 6, 2012 3:47 PM | Permalink | Reply to this

Re: Quivering with Excitement

Some follow up on Quiver Grassmannians and F 1F_1-geometry.

Posted by: David Corfield on May 11, 2012 3:03 PM | Permalink | Reply to this

Re: Quivering with Excitement

I remember liking (stable) Grassmannians because they are classifying spaces for vector bundles. So, my next question would be: do quiver Grassmannians stabilize in a sensible way? If yes, then what does that make a stable algebraic variety? And are they classifying spaces for something?

Posted by: Jesse C. McKeown on May 11, 2012 4:50 PM | Permalink | Reply to this

Re: Quivering with Excitement

Did you realize that the result is already quite old? Have a look to page 31 in

http://arxiv.org/abs/1301.1251

for some of the history.

Posted by: Lutz Hille on June 2, 2013 5:35 PM | Permalink | Reply to this

Re: Quivering with Excitement

There’s been more recent strengthening of this theorem too, that every projective variety is a quiver Grassmannian of any wild acyclic quiver. This result is contained in a five page paper, Quiver Grassmannians for wild acyclic quivers by Ringel (2018).

Posted by: Mike Pierce on March 3, 2020 10:34 AM | Permalink | Reply to this

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