1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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 $Sp_4/N$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.