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24
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1answer
1k 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 ...
24
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1answer
1k 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, ...
17
votes
14answers
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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 ...
14
votes
4answers
2k views

Classification of finite groups of isometries

Consider the problem of classifying the finite groups of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ...
11
votes
2answers
2k views

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 ...
9
votes
1answer
951 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 $G$. ...
4
votes
1answer
181 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 ...
3
votes
0answers
293 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 ...
0
votes
1answer
337 views

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 ...