User yann palu - MathOverflow

Answer by Yann Palu for The category of Abelian groups with selected elements

2011-07-11T12:29:02Z

Here is an easy example which illustrates Francesco Polizzi's answer: Consider the morphisms $\mathbb{Z} \rightarrow\mathbb{Z}/12$ and $\mathbb{Z} \rightarrow\mathbb{Z}/6$, sending 1 to the class of 3. These are objects in the comma category. Now the canonical surjection $\mathbb{Z}/12 \rightarrow \mathbb{Z}/6$ induces a morphism in the comma category (it sends the class of 3 to the class of 3).

Answer by Yann Palu for Books you would like to read (if somebody would just write them...)

2011-01-24T15:14:22Z

"Quiver varieties with a wealth of examples" ?

Answer by Yann Palu for Recommended page layout settings for latex

2011-01-19T16:37:41Z

From arXiv, it is possible to download the source of any paper. You could check the page layout from a paper you like.

Answer by Yann Palu for quiver mutation

2011-01-19T10:31:25Z

First of all, I would like to note that there is a very nice applet, due to Keller, which mutates quivers (and does much more): http://people.math.jussieu.fr/~keller/quivermutation/

Also, many information on cluster algebras (the definition of which requires quiver mutation) can be found at the cluster algebra portal http://www.math.lsa.umich.edu/~fomin/cluster.html

Some very nice introductions and surveys to some of the theories which were developped thanks to cluster algebras and mutation are:

http://people.math.jussieu.fr/~keller/publ/KellerCatAcyclic.pdf

http://people.math.jussieu.fr/~keller/publ/KellerClusterAlgQuivRep.pdf

http://uk.arxiv.org/pdf/1012.4949.pdf

http://uk.arxiv.org/pdf/1012.6014.pdf

http://people.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf

Here are a few examples of areas of research which are related to (or motivated by) Fomin-Zelevinsky's quiver mutation:

Cluster tilting theory (in representation theory of quivers and algebras);

Triangulations of punctured Riemann surfaces;

Higher Teichmuller spaces;

Poisson geometry;

In algebraic geometry: Stability conditions, Calabi-Yau algebras, Donaldson-Thomas invariants...

Let me give a few more details on cluster tilting: The definition of a cluster algebra makes use of the notion of seed mutation. Quiver mutation is a part of this seed mutation. As an analogy, one can consider the flip of triangulations of an n-gone (To flip a triangulation, delete one of its arcs and replace it by the only arc giving a new triangulation). Through this analogy, seeds correspond to triangulations, and seed mutation to flips.

Now, in the representation theory of finite dimensional algebras, there is a notion of tilting modules. Such modules can sometimes be mutated at an indecomposable summand (as triangulations can be flipped at an arc), but not always: somme summands cannot be mutated. Moreover, there is a quiver naturally associated with such a module (the Gabriel quiver of its endomorphism algebra). Through a mutation, the associated quivers are related by Fomin-Zelevinsky's quivers mutation in some cases, but not always.

The whole theory of cluster tilting, including cluster categories and their generalisations, module categories over preprojective algebras, more general Calabi-Yau triangulated categories... arised from the (successful) attempt to fix these two problems in the relation between tilting theory and cluster algebras.

As a concrete application of this theory, on can cite Keller's proof of Zamolodchikov's periodicity conjecture:

http://people.math.jussieu.fr/~keller/publ/KellerPeriodicity.pdf

Answer by Yann Palu for Intuition about the triangulation of a homotopy category K(A)

2010-10-22T08:01:28Z

One possible motivation for considering the triangles in 1. is that they induce long exact sequences
$\cdots \rightarrow Hom_K(Z,C_f[-1]) \rightarrow Hom_K(Z,X) \rightarrow Hom_K(Z,Y)\rightarrow Hom_K(Z,C_f) \rightarrow \cdots$
and
$\cdots \rightarrow Hom_K(C_f,Z) \rightarrow Hom_K(Y,Z)\rightarrow Hom_K(X,Z) \rightarrow Hom_K(C_f[-1],Z) \rightarrow \cdots$
for any $Z$.

I think that the triangulated structure of $K(\mathcal{A})$ reflects (in some sense "up to homotopy") the abelian structure of $C(\mathcal{A})$. Indeed $K(\mathcal{A})$ is the stable category (see the book of D. Happel "Triangulated categories in the representation theory of finite dimensional algebras") associated with the abelian category $C(\mathcal{A})$.

Answer by Yann Palu for Glueing triangulated categories

2010-10-07T14:20:58Z

I'm not sure but Proposition 1.16 in the paper:

http://arxiv.org/pdf/0911.0172

by Iyama-Kato-Miyachi might be related to your question.

Comment by Yann Palu

2011-04-27T11:05:37Z

+1 for the tag cluster-algebras!

Comment by Yann Palu

2011-02-22T15:43:35Z

The reference: "Rump: Almost abelian categories" might interest you. It is proven in particular that if $\mathcal{A}$ is preabelian, then all canonical morphisms $coim(f) \to im(f)$ are mono and epi if and only if the category is semi-abelian (that is kernels and stable under composition and so are cokernels).

Comment by Yann Palu

2011-02-09T09:56:15Z

IF 0 is a cube, then I guess 0,0,0 is a solution...

Comment by Yann Palu

2011-01-19T10:42:52Z

There could/should be a tag: cluster algebras...