Tag Archive for quivers


what have quivers done to students?

A few years ago a student entered my office asking suggestions for his master thesis. “I’m open to any topic as long as it has nothing to do with those silly quivers!” At that time not the best of opening-lines to address me and, inevitably, the most disastrous teacher-student-conversation-ever followed (also on my part, i’m sorry to say). This week,… Read more →


It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma $, or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of… Read more →

quivers versus quilts

We have associated to a subgroup of the modular group $PSL_2(\mathbb{Z}) $ a quiver (that is, an oriented graph). For example, one verifies that the fundamental domain of the subgroup $\Gamma_0(2) $ (an index 3 subgroup) is depicted on the right by the region between the thick lines with the identification of edges as indicated. The associated quiver is then… Read more →

the modular group and superpotentials (2)

Last time we have that that one can represent (the conjugacy class of) a finite index subgroup of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ by a Farey symbol or by a dessin or by its fundamental domain. Today we will associate a quiver to it. For example, the modular group itself is represented by the Farey symbol [tex]\xymatrix{\infty \ar@{-}[r]_{\circ}… Read more →

Superpotentials and Calabi-Yaus

Yesterday, Jan Stienstra gave a talk at theARTS entitled “Quivers, superpotentials and Dimer Models”. He started off by telling that the talk was based on a paper he put on the arXiv Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d’Enfants but that he was not going to say a thing about dessins but would rather focuss on the… Read more →

M-geometry (1)

Take an affine $\mathbb{C} $-algebra A (not necessarily commutative). We will assign to it a strange object called the tangent-quiver $\vec{t}~A $, compute it in a few examples and later show how it connects with existing theory and how it can be used. This series of posts can be seen as the promised notes of my talks at the GAMAP-workshop… Read more →

down with determinants

The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least one eigenvector. Take a vector… Read more →

recap and outlook

After a lengthy spring-break, let us continue with our course on noncommutative geometry and $SL_2(\mathbb{Z}) $-representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification $\widehat{SL_2(\mathbb{Z})} = \underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $ of the modular group $SL_2(\mathbb{Z}) $ and in the knowledge of a certain subgroup G of its group of… Read more →

master class 2007

Next week our master programme on noncommutative geometry will start. Here is the list of all international mini-courses (8 hours each) and firm or tentative dates. For the latest update, it is always best to check with the Arts seminar website. Hans-Juergen Schneider (Munich) “Hopf Galois extensions and quotient theory of Hopf algebras”. February 20-23 each day from 10h30-12h30. Markus… Read more →

coalgebras and non-geometry 3

Last time we saw that the _coalgebra of distributions_ of a noncommutative manifold can be described as a coalgebra Takeuchi-equivalent to the path coalgebra of a huge quiver. This infinite quiver has as its vertices the isomorphism classes of finite dimensional simple representations of the qurve A (the coordinate ring of the noncommutative manifold) and there are as many directed… Read more →

coalgebras and non-geometry

In this series of posts I’ll try to make at least part of the recent [Kontsevich-Soibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a bit more accessible to algebraists. In non-geometry, the algebras corresponding to *smooth affine varieties* I’ll call **qurves** (note that they are called **quasi-free algebras** by Cuntz & Quillen and **formally smooth** by Kontsevich). By definition, a qurve in an affine $\mathbb{C} $-algebra… Read more →

non-(commutative) geometry

Now that my non-geometry post is linked via the comments in this string-coffee-table post which in turn is available through a trackback from the Kontsevich-Soibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connes-style non-commutative geometry is this : if you have a situation where a discrete group is acting with… Read more →


Here’s an appeal to the few people working in Cuntz-Quillen-Kontsevich-whoever noncommutative geometry (the one where smooth affine varieties correspond to quasi-free or formally smooth algebras) : let’s rename our topic and call it non-geometry. I didn’t come up with this term, I heard in from Maxim Kontsevich in a talk he gave a couple of years ago in Antwerp. There… Read more →

why nag? (1)

Let us take a hopeless problem, motivate why something like non-commutative algebraic geometry might help to solve it, and verify whether this promise is kept. Suppose we want to know all solutions in invertible matrices to the braid relation (or Yang-Baxter equation) All such solutions (for varying size of matrices) form an additive Abelian category , so a big step… Read more →