In the webpage, there is a result:

Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $F_4$, $G_2$.

On the other hand, on page 46 of the lecture notes, there is a result:

Theorem 2. A cluster algebra of finite type ifand only if the mutable part of its quiver at some seed is an orientation of a simply-laced Dynkin diagram.

My question is: cluster algebras of finite type corresponds to Dynkin diagrams or only simply-laced Dynkin diagrams? Thank you very much.


You have a finite type cluster algebra associated to every Cartan matrix, regardless whether the Dynkin diagram is simply-laced or not.

The cluster algebras of types $B_n$ and $C_n$ have been studied for example in http://www.dmtcs.org/pdfpapers/dmAJ0138.pdf It is known that the cluster algebra $B_2$ is the coordinate ring of $Sp_4/N$.

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