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Classification of finite type structures in mathematics often lead to the Dynkin diagrams (Example: representation-finite hereditary algebras, simple Lie algebras, Cluster algebras,... and I have read that there are nearly 50 other such structures connected with the Dynkin diagrams).

Questions: Are there classification results of such "finite type" structures where the answer surprisingly does not correspond to exactly the Dynkin diagrams, but to Dynkin diagrams with maybe a finite number of other diagrams and/or some simply laced diagrams missing (like for example $E_6$)?

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    $\begingroup$ This is an interesting, slightly different classification result: arxiv.org/abs/1603.03942 $\endgroup$ – Sam Hopkins Jul 24 '17 at 11:54
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    $\begingroup$ What do you mean here by 'surprisingly'? I would argue that it is the regularity with which Dynkin diagrams are the answer to finite-type classification problems that is surprising! In general I think one should not be surprised if the answer to a classification problem is something else, unless there was a better reason to think the answer might be Dynkin diagrams than merely that you had a finite-type classification problem. $\endgroup$ – Matthew Pressland Jul 24 '17 at 12:14
  • $\begingroup$ Possibly relevant : en.wikipedia.org/wiki/Du_Val_singularity. $\endgroup$ – Watson Jul 25 '17 at 8:49
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How about finite Coxeter groups?

https://en.wikipedia.org/wiki/Coxeter_group

I guess technically this doesn't quite fit since they're classified with Coxeter diagrams not Dynkin diagrams. Essentially you have Dynkin diagrams with the "arrows" removed. Thus, for example, types $B$ and $C$ are the same.

In any case, the classification of finite Coxeter groups yields the standard $A_n$,$B_n=C_n$,$D_n$,$E_6$,$E_7$,$E_8$,$F_4$, and $G_2$ classification (as usual). But it also includes new members $H_2$, $H_3$, $H_4$, and $I_n$.

Jim Humphrey's book "Reflection Groups and Coxeter Groups" is a great resource to learn more about these groups.

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Classification of finite simple groups is an obvious example of the situation where, apart from Dynkin diagrams, one gets few more infinite series of objects (e.g. alternating groups), as well as "true" 26 exceptions.

There is a lot of work done on understanding these 26 exceptions in terms of diagrams; e.g. one would need to consider an extra "type" corresponding to the Petersen graph, or to a finite affine plane, or to a finite design corresponding to one of Mathieu groups.

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You may be interested in the classification of quivers of finite mutation-type: http://www.emis.ams.org/journals/EJC/Volume_15/PDF/v15i1r139.pdf

This extends the classification of mutation classes of quivers determining cluster algebras with only finitely many cluster variables—the answer to this classification problem is given by the simply-laced Dynkin diagrams. (The non-simply-laced diagrams can also be included by defining cluster algebras from skew-symmetrizable, rather than just skew-symmetric, matrices.) These classes of quivers are sometimes said to have finite cluster-type.

All of these mutation classes are necessarily finite (one says the quivers have finite mutation-type), but there are also other quivers with finite mutation-type, including the affine Dynkin diagrams. (In type $\tilde{\mathsf{A}}$ one has to be a bit careful, because not all quivers with this underlying graph are mutation equivalent, and the cyclically oriented $\tilde{A}_n$ quiver is mutation equivalent to $\mathsf{D}_{n+1}$.) One might hope, given that the finite cluster-type problem was solved by Dynkin diagrams, that the slightly more general finite mutation-type problem might be solved by Dynkin + affine diagrams (cf. finite representation-type versus tame representation-type, or positive definite Tits forms versus positive semi-definite Tits forms). However, this is not the case, for several reasons of increasing severity.

Firstly a quiver on two vertices always has finite mutation-type for degeneracy reasons, but it doesn't feel too unreasonable to exclude this case. Next there are quivers determined by triangulations of oriented surfaces with marked points, which are also of finite mutation-type but their mutation classes needn't contain a Dynkin or affine diagram. Finally, Derksen and Weyman show in the paper linked to above that even excluding the previous two cases, there are still some exceptional mutation-finite quivers.

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I would mention the classification of finite-dimensional Nichols algebras $\mathfrak{B}(V_1\oplus\cdots\oplus V_\theta)$, $\theta\geq2$, over decomposable Yetter-Drinfeld modules $V_1\oplus\cdots\oplus V_{\theta}$ over groups.

In the case $\theta\geq4$ the classification is essentially based on Dynking diagrams of finite type (with labels). In the cases $\theta\in\{2,3\}$ there are exceptions, most of them appearing in the case $\theta=2$.

References: https://arxiv.org/abs/1412.0857 and https://arxiv.org/abs/1311.2881.

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