User yann palu - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:13:10Z http://mathoverflow.net/feeds/user/9669 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70008/the-category-of-abelian-groups-with-selected-elements/70010#70010 Answer by Yann Palu for The category of Abelian groups with selected elements Yann Palu 2011-07-11T12:29:02Z 2011-07-11T12:29:02Z <p>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).</p> http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/53070#53070 Answer by Yann Palu for Books you would like to read (if somebody would just write them...) Yann Palu 2011-01-24T15:14:22Z 2011-01-24T15:14:22Z <p>"Quiver varieties with a wealth of examples" ?</p> http://mathoverflow.net/questions/52527/recommended-page-layout-settings-for-latex/52530#52530 Answer by Yann Palu for Recommended page layout settings for latex Yann Palu 2011-01-19T16:37:41Z 2011-01-19T16:37:41Z <p>From arXiv, it is possible to download the source of any paper. You could check the page layout from a paper you like.</p> http://mathoverflow.net/questions/36536/quiver-mutation/52501#52501 Answer by Yann Palu for quiver mutation Yann Palu 2011-01-19T10:31:25Z 2011-01-19T10:41:03Z <p>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): <a href="http://people.math.jussieu.fr/~keller/quivermutation/" rel="nofollow">http://people.math.jussieu.fr/~keller/quivermutation/</a></p> <p>Also, many information on cluster algebras (the definition of which requires quiver mutation) can be found at the cluster algebra portal <a href="http://www.math.lsa.umich.edu/~fomin/cluster.html" rel="nofollow">http://www.math.lsa.umich.edu/~fomin/cluster.html</a></p> <p>Some very nice introductions and surveys to some of the theories which were developped thanks to cluster algebras and mutation are:</p> <p><a href="http://people.math.jussieu.fr/~keller/publ/KellerCatAcyclic.pdf" rel="nofollow">http://people.math.jussieu.fr/~keller/publ/KellerCatAcyclic.pdf</a></p> <p><a href="http://people.math.jussieu.fr/~keller/publ/KellerClusterAlgQuivRep.pdf" rel="nofollow">http://people.math.jussieu.fr/~keller/publ/KellerClusterAlgQuivRep.pdf</a></p> <p><a href="http://uk.arxiv.org/pdf/1012.4949.pdf" rel="nofollow">http://uk.arxiv.org/pdf/1012.4949.pdf</a></p> <p><a href="http://uk.arxiv.org/pdf/1012.6014.pdf" rel="nofollow">http://uk.arxiv.org/pdf/1012.6014.pdf</a></p> <p><a href="http://people.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf" rel="nofollow">http://people.math.jussieu.fr/~keller/publ/KellerCYtriangCat.pdf</a></p> <p>Here are a few examples of areas of research which are related to (or motivated by) Fomin-Zelevinsky's quiver mutation:</p> <p>Cluster tilting theory (in representation theory of quivers and algebras);</p> <p>Triangulations of punctured Riemann surfaces;</p> <p>Higher Teichmuller spaces;</p> <p>Poisson geometry;</p> <p>In algebraic geometry: Stability conditions, Calabi-Yau algebras, Donaldson-Thomas invariants...</p> <p>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.</p> <p>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.</p> <p>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. </p> <p>As a concrete application of this theory, on can cite Keller's proof of Zamolodchikov's periodicity conjecture:</p> <p><a href="http://people.math.jussieu.fr/~keller/publ/KellerPeriodicity.pdf" rel="nofollow">http://people.math.jussieu.fr/~keller/publ/KellerPeriodicity.pdf</a></p> http://mathoverflow.net/questions/43107/intuition-about-the-triangulation-of-a-homotopy-category-ka/43145#43145 Answer by Yann Palu for Intuition about the triangulation of a homotopy category K(A) Yann Palu 2010-10-22T08:01:28Z 2010-10-22T10:45:37Z <p>One possible motivation for considering the triangles in 1. is that they induce long exact sequences<br> $\cdots \rightarrow Hom_K(Z,C_f[-1]) \rightarrow Hom_K(Z,X) \rightarrow Hom_K(Z,Y)\rightarrow Hom_K(Z,C_f) \rightarrow \cdots$<br> and<br> $\cdots \rightarrow Hom_K(C_f,Z) \rightarrow Hom_K(Y,Z)\rightarrow Hom_K(X,Z) \rightarrow Hom_K(C_f[-1],Z) \rightarrow \cdots$<br> for any $Z$.</p> <p>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 $C(\mathcal{A})$.</p> http://mathoverflow.net/questions/41416/glueing-triangulated-categories/41419#41419 Answer by Yann Palu for Glueing triangulated categories Yann Palu 2010-10-07T14:20:58Z 2010-10-07T14:20:58Z <p>I'm not sure but Proposition 1.16 in the paper:</p> <p><a href="http://arxiv.org/pdf/0911.0172" rel="nofollow">http://arxiv.org/pdf/0911.0172</a></p> <p>by Iyama-Kato-Miyachi might be related to your question.</p> http://mathoverflow.net/questions/63070/which-cluster-algebras-are-coordinate-rings-of-double-bruhat-cells Comment by Yann Palu Yann Palu 2011-04-27T11:05:37Z 2011-04-27T11:05:37Z +1 for the tag cluster-algebras! http://mathoverflow.net/questions/41722/is-every-balanced-pre-abelian-category-abelian Comment by Yann Palu Yann Palu 2011-02-22T15:43:35Z 2011-02-22T15:43:35Z The reference: &quot;Rump: Almost abelian categories&quot; 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). http://mathoverflow.net/questions/54862/230-april-1915-proposed-by-e-b-escott-ann-arbor-michigan Comment by Yann Palu Yann Palu 2011-02-09T09:56:15Z 2011-02-09T09:56:15Z IF 0 is a cube, then I guess 0,0,0 is a solution... http://mathoverflow.net/questions/36536/quiver-mutation/52501#52501 Comment by Yann Palu Yann Palu 2011-01-19T10:42:52Z 2011-01-19T10:42:52Z There could/should be a tag: cluster algebras...