TITLE: Classification of finite type structures leads to Dynkin diagrams?
QUESTION [14 upvotes]: 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 $E_6$)?

REPLY [7 votes]: 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.