Questions tagged [ade-classifications]

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Felix Schuller's proof of the tautness of rational singularities

I'm reading this paper written by Felix Schuller. https://docserv.uni-duesseldorf.de/servlets/DerivateServlet/Derivate-24686/singularities-bib.pdf Corollary 5.7 writes A rational double point is taut ...
George's user avatar
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3 votes
1 answer
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Classification of "homogeneous" submanifolds of ℝⁿ

I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if: it is a closed connected smooth submanifold of $\mathbb R^n$, for every $p, q$ in $M$, there is a ...
Andrea Aveni's user avatar
3 votes
0 answers
97 views

p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some ...
Roger Van Peski's user avatar
0 votes
1 answer
160 views

dimension vector of indecomposable module over preprojective algebra

It is well-known that there are finitely many indecomposable module over the preprojective algebra associated to a quiver $Q$ if and only if $Q=A_2,A_3,A_4$ and tame type for $A_5$ and wild for others....
Ben's user avatar
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11 votes
1 answer
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equivariant stable class of quaternionic Hopf fibration in RO(G)-degrees of ADE-type

Does the quaternionic Hopf fibration possibly represent a non-torsion element in the $G$-equivariant stable homotopy groups of spheres, for $G$ a finite subgroup of $SO(3)$ and in RO(G)-degree being ...
Urs Schreiber's user avatar
10 votes
2 answers
343 views

Symmetries of module categories over the category of representations of quantum $sl(2)$

The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...
Daniel Whalen's user avatar
4 votes
1 answer
461 views

Kodaira classification and the McKay correspondence

Kodaira's table of singular fibers has a singular fiber for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected ...
Mohammad Farajzadeh-Tehrani's user avatar
3 votes
0 answers
390 views

Blowing up a projective surface

Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...
Naga Venkata's user avatar
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10 votes
1 answer
1k views

Du Val singularity of type G=A,D,E and "small" representations of G

We all know that a simple singularity $W_G(x_1,x_2,x_3)=0$ of type G=A,D,E has the following nice deformation involving the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$. ...
Yuji Tachikawa's user avatar
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1 answer
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What is your favorite ADE-style classification? [duplicate]

Possible Duplicate: ADE type Dynkin diagrams What is your favorite ADE-style classification? Here ADE style is to be understood in a very broad sense. A classification which is not precisely ...
12 votes
2 answers
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Integral positive definite quadratic forms and graphs

Let me start with a question for which I know the answer. Consider a symmetric integral $n\times n$ matrix $A=(a_{ij})$ such that $a_{ii}=2$, and for $i\ne j$ one has $a_{ij}=0$ or $-1$. One can ...
VA.'s user avatar
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32 votes
4 answers
6k views

Classification of finite groups of isometries

Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$. For $n=2$ it is cyclic and dihedral groups. For $n=3$ they are well known, probably from Kepler and are related ...
Mathieu Dutour Sikiric's user avatar
33 votes
1 answer
3k views

Is there a common genesis for ADE classifications?

Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one ...
Paul Siegel's user avatar
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26 votes
1 answer
2k views

Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
Noah Snyder's user avatar
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34 votes
17 answers
3k views

ADE type Dynkin diagrams

The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are: (1) Classification of simply laced complex simple lie algebras. (2) Finite subgroups ...
Avan Thiyagarajan's user avatar