
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 openinglines to address me and, inevitably, the most disastrous teacherstudentconversationever followed (also on my part, i’m… Read more »

The Monster is the largest of the 26 sporadic simple groups and has order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71. It is not so much the size… 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… Read more »

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…. Read more »

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… Read more »

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… Read more »

Take an affine $\mathbb{C} $algebra A (not necessarily commutative). We will assign to it a strange object called the tangentquiver $\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… Read more »

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… Read more »

After a lengthy springbreak, 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… Read more »

Next week our master programme on noncommutative geometry will start. Here is the list of all international minicourses (8 hours each) and firm or tentative dates. For the latest update, it is always best to check with the Arts seminar website. HansJuergen Schneider (Munich) “Hopf Galois extensions and quotient theory of Hopf algebras”. February 2023… Read more »

Last time we saw that the _coalgebra of distributions_ of a noncommutative manifold can be described as a coalgebra Takeuchiequivalent 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… Read more »

In this series of posts I’ll try to make at least part of the recent [KontsevichSoibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a bit more accessible to algebraists. In nongeometry, the algebras corresponding to *smooth affine varieties* I’ll call **qurves** (note that they are called **quasifree algebras** by Cuntz & Quillen and **formally smooth** by Kontsevich). By definition, a qurve… Read more »

Now that my nongeometry post is linked via the comments in this stringcoffeetable post which in turn is available through a trackback from the KontsevichSoibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connesstyle noncommutative geometry is this : if you have a situation where a… Read more »

Here’s an appeal to the few people working in CuntzQuillenKontsevichwhoever noncommutative geometry (the one where smooth affine varieties correspond to quasifree or formally smooth algebras) : let’s rename our topic and call it nongeometry. I didn’t come up with this term, I heard in from Maxim Kontsevich in a talk he gave a couple of… Read more »

Let us take a hopeless problem, motivate why something like noncommutative 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 YangBaxter equation) All such solutions (for varying size of matrices) form an additive Abelian category… Read more »

For finite dimensional hereditary algebras, one can describe its noncommutative topology (as developed in part 2) explicitly, using results of Markus Reineke in The monoid of families of quiver representations. Consider a concrete example, say $A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C} \end{bmatrix}$ where $V$ is an ndimensional complex vectorspace, or equivalently,… Read more »

A *qurve* is an affine algebra such that $~\Omega^1~A$ is a projective $~A~$bimodule. Alternatively, it is an affine algebra allowing lifts of algebra morphisms through nilpotent ideals and as such it is the ‘right’ noncommutative generalization of Grothendieck’s smoothness criterium. Examples of qurves include : semisimple algebras, coordinate rings of affine smooth curves, hereditary orders… Read more »

Here the story of an idea to construct new examples of noncommutative compact manifolds, the computational difficulties one runs into and, when they are solved, the white noise one gets. But, perhaps, someone else can spot a gem among all gibberish… [Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasifree algebras, aka formally smooth algebras) are the \’affine\’ pieces of noncommutative… Read more »

Some people objected to the setup of TheLibrary because it was serving only onepage at a time. They’d rather have a longer downloadtime if they can then browse through the paper/book, download it and print if they decide to do so. Fine! Today I spend some hours refilling TheLibrary with texts. As before you are… Read more »

I expect to be writing a lot in the coming months. To start, after having given the course once I noticed that I included a lot of new material during the talks (mainly concerning the component coalgebra and some extras on noncommutative differential forms and symplectic forms) so I\’d better update the Granada notes soon… Read more »

I have been posting before on the necklace Lie algebra : on Travis Schedler's extension of the Lie algebra structure to a Lie bialgebra and its deformation and more recently in connection with Michel Van den Bergh's double Poisson paper. Yesterday, Victor Ginzburg and Travis Schedler posted their paper Moyal quantization of necklace Lie algebras… Read more »

Tomorrow I’ll give my last class of the semester (year?) so it is about time to think about things to do (such as preparing the courses for the “master program on noncommutative geometry”) and changes to make to this weblog (now that it passed the 25000 mark it is time for something different). In the… Read more »

[Last time][1] we saw that the algebra $(\Omega_V~C Q,Circ)$ of relative differential forms and equipped with the Fedosov product is again the path algebra of a quiver $\tilde{Q}$ obtained by doubling up the arrows of $Q$. In our basic example the algebra map $C \tilde{Q} \rightarrow \Omega_V~C Q$ is clarified by the following picture of… Read more »

The previous post in this sequence was [moduli spaces][1]. Why did we spend time explaining the connection of the quiver $Q~:~\xymatrix{\vtx{} \ar[rr]^a & & \vtx{} \ar@(ur,dr)^x} $ to moduli spaces of vectorbundles on curves and moduli spaces of linear control systems? At the start I said we would concentrate on its _double quiver_ $\tilde{Q}~:~\xymatrix{\vtx{} \ar@/^/[rr]^a… Read more »

In [the previous part][1] we saw that moduli spaces of suitable representations of the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{} \ar@(ur,dr)} $ locally determine the moduli spaces of vectorbundles over smooth projective curves. There is yet another classical problem related to this quiver (which also illustrates the idea of looking at families of moduli spaces… Read more »
Close