Gustavo Jasso

Example 5. Let p = (3, 3, 3) so that
L = 〈⃗x, ⃗y, ⃗z | 3⃗x = 3⃗y = 3⃗z =: ⃗c〉
(we do not need to worry about λ in this particular case). Consider the canonical configuration
T of coh X, c.f. Proposition 2. Then Λ = End X (T ) is given by the quiver
O(⃗x)
x 2
O(2⃗x)
O(⃗y)
y 2
x 1
y 3
O(2⃗y)
x 3
O
O(⃗c)
y 1
z 3
z 1
O(⃗z)
z 2
O(2⃗z)
subject to the relation x 3 + y 3 + z 3 = 0. As explained in Section 2.3, the cluster-tilted
algebra End C (T ) ∼ = Π 3 (Λ) is given by the Jacobian algebra of the graded quiver
O(⃗x)
x 2
O(2⃗x)
O(⃗y)
y 2
x 1
y 3
O(2⃗y)
x 3
O
O(⃗c)
y 1
z 3
ξ
z 1
O(⃗z)
z 2
O(2⃗z)
with potential
W = (x 3 + y 3 + z 3 )ξ
where the only arrow of degree 1 is colored gray. It is easy to see that O is a formal
source of T since a relation in End X (T ) begins at O, c.f. [8, Kor. 3.5]. Then the algebra
End C (µ O T ) is given by the Jacobian algebra of the graded quiver
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