ADE type Dynkin diagrams - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:53:06Z http://mathoverflow.net/feeds/question/6781 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams ADE type Dynkin diagrams Avan Thiyagarajan 2009-11-25T10:25:27Z 2010-07-26T19:03:09Z <p>The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are:</p> <p>(1) Classification of simply laced complex simple lie algebras.</p> <p>(2) Finite subgroups of $Sl_2 (\mathbb{C})$</p> <p>Are there any other objects that they classify?</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6784#6784 Answer by David Corfield for ADE type Dynkin diagrams David Corfield 2009-11-25T10:46:46Z 2009-11-25T10:46:46Z <p>Have you read <a href="http://math.ucr.edu/home/baez/ADE.html" rel="nofollow">this</a> by John McKay and the 6 editions of This Week's Finds listed at the end? <a href="http://math.ucr.edu/home/baez/week230.html" rel="nofollow">Week 230</a> gives plenty of ADE appearances.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6787#6787 Answer by Gray Taylor for ADE type Dynkin diagrams Gray Taylor 2009-11-25T11:24:24Z 2009-11-25T11:24:24Z <p>Let $G$ be a connected graph with the property that all eigenvalues of $G$ lie in $[-2,2]$ (such a $G$ is called cyclotomic). Then $G$ is either one of $\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$, an $\tilde{A}_n$ for $n\ge 2$, a $\tilde{D}_n$ for $n\ge4$, or an induced subgraph of one of these. In other words, the ADE graphs classify the maximal cyclotomic graphs. </p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6791#6791 Answer by Mariano Suárez-Alvarez for ADE type Dynkin diagrams Mariano Suárez-Alvarez 2009-11-25T12:02:13Z 2009-11-25T12:23:07Z <p>They classify quivers for which the path algebra is of finite representation type., acording to a famous theorem of P. Gabriel.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6792#6792 Answer by Mariano Suárez-Alvarez for ADE type Dynkin diagrams Mariano Suárez-Alvarez 2009-11-25T12:03:23Z 2009-11-25T12:03:23Z <p>They parametrize the finite dimensional preprojective algebras.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6793#6793 Answer by Mariano Suárez-Alvarez for ADE type Dynkin diagrams Mariano Suárez-Alvarez 2009-11-25T12:13:25Z 2009-11-25T12:29:44Z <p>They classify the so called "simple singularities" of differential maps, that is, of those types of singularities which involve no parameters. See V. I. Arnolʹd, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. 2, Chap. 15, sect. 1.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6794#6794 Answer by Mariano Suárez-Alvarez for ADE type Dynkin diagrams Mariano Suárez-Alvarez 2009-11-25T12:22:30Z 2009-11-25T12:41:28Z <p>They classify the cluster algebras of finite type (that is, with a finite number of clusters). See S. Fomin, A. Zelevinski, <em>Cluster algebras II: Finite type classification</em>, Invent. Math. 154 (2003), 63--121.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6797#6797 Answer by Mariano Suárez-Alvarez for ADE type Dynkin diagrams Mariano Suárez-Alvarez 2009-11-25T12:37:41Z 2009-11-25T12:37:41Z <p>They classify the (germs of) isolated rational singular points of two dimensional complex analytic spaces. See A. E. Durfee, <em>Fifteen characterizations of rational double points and simple critical points</em>. Enseign. Math. (2) 25 (1979), no. 1-2, 131--163.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6801#6801 Answer by Dave Penneys for ADE type Dynkin diagrams Dave Penneys 2009-11-25T13:05:52Z 2009-11-25T13:05:52Z <p>They classify principal graphs of $II_1$-subfactors with index less than $4$. The principal graph can be $A_n$, $D_{2n}$, $E_6$, or $E_8$, but $D_{odd}$ and $E_7$ do not occur.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6803#6803 Answer by José Figueroa-O'Farrill for ADE type Dynkin diagrams José Figueroa-O'Farrill 2009-11-25T13:06:26Z 2009-11-25T13:06:26Z <p>They classify certain types of rational conformal field theories, as in this <a href="http://arxiv.org/abs/0911.3242" rel="nofollow">recent review paper</a>.</p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/6804#6804 Answer by Thomas Riepe for ADE type Dynkin diagrams Thomas Riepe 2009-11-25T13:09:06Z 2009-11-25T13:09:06Z <p><a href="http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf" rel="nofollow" title="scan">This article</a> gives a nice overview on the "ADE-problem". It is written in the late 1970's, so it does not cover more recent appearences. Where is their appearance is most surprising? </p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/17053#17053 Answer by Helene Tyler for ADE type Dynkin diagrams Helene Tyler 2010-03-04T04:15:35Z 2010-03-04T04:15:35Z <p>As Mariano said, the ADE Dynkin diagrams classify quivers of finite representation type. But wait, there's more. If you add one more vertex to a Dynkin diagram (in a particular way, not an arbitrary one), you get an extended Dynkin diagram (aka a Euclidian diagram). The extended ADE diagrams classify quivers of tame representation type. This is related to the fact that the extended ADE diagrams give you a positive semi-definite Tits form, while the ordinary ADE diagrams give you a positive definite Tits form. </p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/26217#26217 Answer by Suren Fernando for ADE type Dynkin diagrams Suren Fernando 2010-05-28T01:20:26Z 2010-05-28T01:20:26Z <p>You might also take a look at Slodowy, P. (1983), Platonic Solids, Kleinian Singularities and Lie Groups,Lecture Notes in Mathematics, No. 1008, pp. 102- 138, </p> <p>Arnold's Trinities paper "Polymathematics : is mathematics a single science or a set of arts?", easily found on the net </p> <p>and </p> <p>Chapoton's own trinities page (also found on the net). </p> <p>The last two focus on the "E" part of ADE, but give long lists of intriguing parallels. </p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/31978#31978 Answer by JME for ADE type Dynkin diagrams JME 2010-07-15T08:41:51Z 2010-07-26T19:03:09Z <p>Extended Dynkin diagrams appear naturally in Kodaira classification of singular fibers of an elliptic surface. </p> http://mathoverflow.net/questions/6781/ade-type-dynkin-diagrams/33422#33422 Answer by Petya for ADE type Dynkin diagrams Petya 2010-07-26T17:43:10Z 2010-07-26T17:43:10Z <p>I suggest to take a look on a very nice Givental's paper (MR1138519 (92k:58031)):</p> <p>"Reflection groups in singularity theory." </p> <p>Here is the review (by V.D. Sedykh): The simple singularities of functions are classified by the Coxeter groups $(A,D,E$-classification). This classification arises in other problems, too (the classifications of the simple Lie algebras, of the finite quaternions groups and so on). The author gives a detailed survey of these results. He also considers the problems connected with the classification of the quasihomogeneous unimodular singularities of functions (the classification of the degenerations of elliptic curves, the theory of automorphic functions and so on) in this paper.</p>