Proceedings of the 44th Symposium on Ring Theory and ...

Consider <strong>the</strong> associated QP (Q Λ , W Λ , C Λ ) <strong>of</strong> Λ and we put ˜µ k (Q Λ , W Λ ) = (Q ′ , W ′ ).
Then W ′ is given by
W ′ = [ ∑ ∑
r r] + [ρ r a]a
r∈Rρ ∗ ρ ∗ r,
and it is easy to check that subset
a∈Q 1 ,r∈R
s(a)=k=s(r)
C ′ = { ρ r | r ∈ R, s(r) ≠ k} ∐ { [ρ r a] | a ∈ Q 1 , r ∈ R, s(a) = k = s(r)}
<strong>of</strong> Q ′ is a cut <strong>of</strong> (Q ′ , W ′ ).
Then we have <strong>the</strong> following.
Theorem 7. Let Λ = ̂KQ/〈R〉 be a finite dimensional algebra with a minimal set <strong>of</strong>
relations. Let T k := τ − P k ⊕ Λ/P k be <strong>the</strong> APR tilting module. Then if inj.dimP k ≤ 2, we
have an algebra isomorphism
End Λ (T k ) ∼ = P(˜µ k (Q Λ , W Λ ), C ′ ).
Notice that <strong>the</strong> assumption inj.dimP k ≤ 2 is automatic if gl.dimΛ = 2. Thus our
<strong>the</strong>orem give a generalization from gl.dimΛ = 1 to gl.dimΛ = 2.
Here we will explain <strong>the</strong> choice <strong>of</strong> C ′ . In fact C ′ is naturally obtained by using graded
mutations. For this purpose, we recall graded QPs, as introduced by [1].
Graded quivers with potentials. Let (Q, W ) be a QP and we define a map d : Q 1 → Z.
We call a QP (Q, W, d) Z-graded QP if each arrow a ∈ Q 1 has a degree d(a) ∈ Z, and
homogeneous <strong>of</strong> degree l if each term in W is a degree l.
Definition 8. Let QP (Q, W, d) be a Z-graded QP <strong>of</strong> degree l. For each vertex k in Q
not lying on a loop nor 2-cycle, we define a left mutation µ L k (Q, W, d) as a reduced part
<strong>of</strong> ˜µ L k (Q, W, d) = (Q′ , W ′ , d ′ ), where (Q ′ , W ′ , d ′ ) is given as follows.
(1) (Q ′ , W ′ ) = ˜µ k (Q, W )
(2) The new degree d ′ is defined as follows:
• d ′ (a) = d(a) for each arrow a ∈ Q ∩ Q ′ .
• d ′ (a ∗ ) = −d(a) for each arrow a : k → v in Q.
• d ′ (b ∗ ) = −d(b) + l for each arrow b : u → k in Q.
• d ′ ([ba]) = d(a) + d(b) for each pair <strong>of</strong> arrows u → b k → a v in Q.
In particular, ˜µ L k (Q, W, d) also has a potential <strong>of</strong> degree l. Similarly, we can define
˜µ R k at k. In this case, we define d′ (b ∗ ) = −d(b) for each arrow b : u → k in Q and
d ′ (a ∗ ) = −d(a) + l for each arrow a : k → v in Q.
If (Q, W ) has a cut C, we can identify <strong>the</strong> QP with a Z-graded QP <strong>of</strong> degree 1 associating
a grading on Q by
{
1 a ∈ C
d C (a) =
0 a ∉ C.
We denote by (Q, W, C) <strong>the</strong> graded QP <strong>of</strong> degree 1 with this grading. If any arrow <strong>of</strong>
˜µ L k (Q, W, C) has degree 0 or 1, degree 1 arrows give a cut <strong>of</strong> ˜µ k(Q, W ) since ˜µ L k (Q, W, C)
is homogeneous <strong>of</strong> degree 1. Therefore a cut <strong>of</strong> ˜µ k (Q Λ , W Λ ) is naturally induced as degree
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