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...on-frozen vertex i P Q0 is called green ifsj1 P F | D iÝÑ j1 P Q1 ( “ H. It is called red ifsj1 P F | D j1ÝÑ i P Q1 ( “ H. We warn the reader that we have adopted the conventions opposite to those in =-=[16, 65]-=- to keep coherent with mutations of the categorical objects. When we mutate at a green or red vertex, the above step (4) is redundant because in this case no arrows between frozen vertices are produce...

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...it produces a quiver which does not contain such a vertex i. Maximal green sequences have been introduced in [K]. They are related to quantum dilogarithm identities, DT invariants and BPS states, see =-=[BDP]-=- and the references therein. The question whether or not there exists a quiver in the mutation class of Q which admits a maximal green sequence seems related to the question whether or not A(Q) = U(Q)...

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...g at any vertex of Q; • The cluster algebra A(Q) is not Noetherian [24]; • The upper cluster algebra U(Q) is not equal to the cluster algebra A(Q) [3, 24]; • Q has no maximal green mutation sequences =-=[5]-=-; • There is a non-degenerate potential on Q defining a Hom-finite cluster category having cluster-tilting objects that are not reachable from the canonical one [25]; • Q does not belong to the class ...

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... Bt), furthermore ct is a basis of Zn [10, Proposition 1.3]. We also write cj > 0 (resp. cj < 0) if all entries are non-negative (resp. non-positive). Now we can recall the notion of a green sequence =-=[3]-=-: Definition 1. Let B0 be a skew-symmetrizable n × n matrix. A green sequence for B0 is a sequence i = (i1, . . . , il) such that, for any 1 ≤ k ≤ l with (c, B) = µik−1 ◦ · · · ◦ µi1(c0, B0), we have ...