Xueyu Luo

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is isomorphic to the following R-order: ⎡ R R R R (x −1 ⎤ ) R R (x) R R R R R R P(Q, W, F ) ∼ (x 2 ) (x) R R R (x) (x) = (x 2 ) (x 2 ) (x) R R (x) (x) ⎢(x 2 ) (x 2 ) (x 2 ) (x) R (x 2 ) (x) ⎥ ⎣ (x) (x) R R R R R ⎦ (x) (x) (x) R R (x) R It is clear that e F P(Q, W, F )e F ∼ = Λ5 holds in this case. The Auslander-Reiten quiver of Λ 5 is the following: ⎡ ⎤ R (x) ⎢(x 2 ) ⎥ ⎣(x 2 ) ⎦ (x 2 ) ⎡ ⎤ R R ⎢ R ⎥ ⎣(x) ⎦ (x) ⎡ ⎤ R (x) ⎢ (x) ⎥ ⎣(x 2 ) ⎦ (x 2 ) ⎡ ⎤ R R ⎢ R ⎥ ⎣ R ⎦ (x) ⎡ ⎤ R R ⎢ (x) ⎥ ⎣(x 2 ) ⎦ (x 2 ) ⎡ ⎤ R (x) ⎢ (x) ⎥ ⎣ (x) ⎦ (x 2 ) ⎡ ⎤ R R ⎢ (x) ⎥ ⎣ (x) ⎦ (x 2 ) ⎡ ⎤ R (x) ⎢(x) ⎥ ⎣(x) ⎦ (x) ⎡ ⎢ ⎣ R R R (x) (x 2 ) ⎤ ⎥ ⎦ ⎡ ⎤ R R ⎢(x) ⎥ ⎣(x) ⎦ (x) ⎡ ⎤ R R ⎢ R ⎥ ⎣(x) ⎦ (x) ⎡ ⎤ R (x) ⎢(x 2 ) ⎥ ⎣(x 2 ) ⎦ (x 2 ) As a Λ 5 -module, e F P(Q, W, F ) is isomorphic to ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ R (x) (x 2 ) (x 2 ) (x 2 ) ⎢ ⎣ ⎥ ⎢ ⎦ ⊕ ⎣ R R (x) (x 2 ) (x 2 ) ⎥ ⎢ ⎦ ⊕ ⎣ R R R (x) (x 2 ) ⎥ ⎢ ⎦ ⊕ ⎣ R R R R (x) ⎥ ⎦ ⊕ ⎡ R (x) (x) (x) (x) ⎢ ⎣ ⎤ ⎡ ⎥ ⎢ ⎦ ⊕ ⎣ ⎤ R R (x) (x) (x) ⎡ ⎥ ⎢ ⎦ ⊕ ⎣ which is cluster tilting in CM(Λ 5 ). As expected, its summands correspond bijectively to the (internal and external) edges of the triangulation, or equivalently to the vertices of the corresponding quiver. –97– R R (x) (x) (x 2 ) ⎤ ⎥ ⎦
The stable category CM(Λ 5 )isequivalenttoC A2 and has a cluster tilting object ⎡ ⎤ ⎡ ⎤ R R (x) (x) (x) ⎢ ⎣ ⎥ ⎢ ⎦ ⊕ ⎣ R R (x) (x) (x 2 ) ⎥ ⎦. References [1] A. B. Buan, O. Iyama, I. Reiten, and D. Smith. Mutation of cluster-tilting objects and potentials. Amer. J. Math., 133(4):835–887, 2011. [2] P. Caldero, F. Chapoton, and R. Schiffler. Quivers with relations arising from clusters (A n case). Trans. Amer. Math. Soc., 358(3):1347–1364, 2006. [3] Sergey Fomin, Michael Shapiro, and Dylan Thurston. Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math., 201(1):83–146, 2008. [4] Daniel Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3), 98(3):797–839, 2009. Graduate School of Mathematics Nagoya University Furocho, Chikusaku, Nagoya 464-8602 JAPAN E-mail address: d1005p@math.nagoya-u.ac.jp –98–
Page 1 and 2: REALIZING CLUSTER CATEGORIES OF DYN
Page 3 and 4: Letusfixapostiveintegern ≥ 3 and
Page 5: C i := C 1 i = C 2 i holds in P(Q,

Xueyu Luo

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