
The natural habitat of this lesson is a bit further down the course, but it was called into existence by a comment/question by Kea I don’t yet quite see where the nc manifolds are, but I guess that’s coming. As I’m enjoying telling about all sorts of sources of finite dimensional representations of $SL_2(\mathbb{Z}) $… 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 »

Last time we have seen that the _coalgebra of distributions_ of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form $\mathbb{C}[[ x_1,\ldots,x_d ]] $ where $d $ is the dimension of the variety. Generalizing this to _noncommutative_ manifolds, the… 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 »

For a qurve (aka formally smooth algebra) A a *block* is a (possibly infinite dimensional over the basefield) left Amodule X such that its endomorphism algebra $D = End_A(X)$ is a division algebra and X (considered as a right Dmodule) is finite dimensional over D. If a block X is finite dimensional over the basefield,… 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 »

Klein’s quartic $X$is the smooth plane projective curve defined by $x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical objects around. For example, it is a Hurwitz curve meaning that the finite group of symmetries (when the genus is at least two this group can have at most $84(g1)$ elements) is as large as possible,… 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 »

I found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front cover the list of the 26 _known_ sporadic groups together with a starred added in proof saying added in proof … the classification of finite simple groups is complete. there are no other sporadic groups. (click… Read more »

Last time we have seen that in order to classify all noncommutative $l$points one needs to control the finite dimensional simple algebras having as their center a finite dimensional fieldextension of $l$. We have seen that the equivalence classes of simple algebras with the same center $L$ form an Abelian group, the Brauer group. The… Read more »

Noncommutative geometry seems pretty trivial compared to commutative geometry : there are just two types of manifolds, points and curves. However, nobody knows how to start classifying these noncommutative curves. I do have a conjecture that any noncommutative curve can (up to noncommutative birationality) be constructed from hereditary orders over commutative curves by universal methods… Read more »

Never ever tell an ICTaware person that you want to try to set up a homenetwork before you understand all 65536 portnumbers and their corresponding protocols. Here is what happened to me this week. Jan Adriaenssens returned from an extended vacation in New Zealand and I told him about my problems with trying to set… Read more »
Close