Quivers with potentials and their representations. I: Mutations.

*(English)*Zbl 1204.16008Mutations of quivers with potentials are introduced and studied. Fix a field \(K\) and a finite quiver \(Q\). A potential can be understood as a countable combination of cyclic paths in \(Q\), that is, an element of the complete path \(K\)-algebra of \(Q\). For a potential \(S\) the Jacobian ideal \(J(S)\) is defined as the closure of the two-sided ideal generated by all partial derivatives of \(S\) with respect to the arrows. The closure is considered with respect to the \(\mathfrak m\)-adic topology, where \(\mathfrak m\) is the ideal generated by all arrows. The factor of the completed path algebra of \(Q\) modulo \(J(S)\) is called the Jacobian algebra.

A quiver with potential (QP) is a quiver \(Q\) that has no loops, equipped with a potential \(S\) such that no cyclically equivalent cyclic paths appear in the decomposition of \(S\). Cyclic equivalence is the equality modulo the \(\mathfrak m\)-adic closure of the space of all commutators in the complete path algebra. If \((Q,S)\) is a QP and a vertex \(k\) does not belong to an oriented cycle of length 2 then the mutation of \(\widetilde\mu_k(Q,S)\) of \((Q,S)\) is defined.

Further, decorated representations of quivers with potentials and their mutations are introduced. It turns out that the construction extends directly the Bernstein-Gelfand-Ponomarev reflection functors and their version for decorated representations studied by R. Marsh, M. Reineke and A. Zelevinsky [in Trans. Am. Math. Soc. 355, No. 10, 4171-4186 (2003; Zbl 1042.52007)]. The construction is an analogue of the one discussed by O. Iyama and I. Reiten, [in Am. J. Math. 130, No. 4, 1087-1149 (2008; Zbl 1162.16007)], in the context of Calabi-Yau algebras. One of the results of the paper asserts that if the quiver \(Q\) is mutation equivalent to a Dynkin quiver then the Jacobson algebra associated with \(Q\) and an (explicitly given) potential \(S\) is isomorphic to the cluster-tilted algebra associated to \(Q\), see A. B. Buan, R. J. Marsh, I. Reiten [Trans. Am. Math. Soc. 359, No. 1, 323-332 (2007; Zbl 1123.16009)].

The article is to be continued in part II [J. Am. Math. Soc. 23, No. 3, 749-790 (2010; Zbl 1208.16017)] on applications of representations of quivers with potentials and their mutations to cluster algebras.

A quiver with potential (QP) is a quiver \(Q\) that has no loops, equipped with a potential \(S\) such that no cyclically equivalent cyclic paths appear in the decomposition of \(S\). Cyclic equivalence is the equality modulo the \(\mathfrak m\)-adic closure of the space of all commutators in the complete path algebra. If \((Q,S)\) is a QP and a vertex \(k\) does not belong to an oriented cycle of length 2 then the mutation of \(\widetilde\mu_k(Q,S)\) of \((Q,S)\) is defined.

Further, decorated representations of quivers with potentials and their mutations are introduced. It turns out that the construction extends directly the Bernstein-Gelfand-Ponomarev reflection functors and their version for decorated representations studied by R. Marsh, M. Reineke and A. Zelevinsky [in Trans. Am. Math. Soc. 355, No. 10, 4171-4186 (2003; Zbl 1042.52007)]. The construction is an analogue of the one discussed by O. Iyama and I. Reiten, [in Am. J. Math. 130, No. 4, 1087-1149 (2008; Zbl 1162.16007)], in the context of Calabi-Yau algebras. One of the results of the paper asserts that if the quiver \(Q\) is mutation equivalent to a Dynkin quiver then the Jacobson algebra associated with \(Q\) and an (explicitly given) potential \(S\) is isomorphic to the cluster-tilted algebra associated to \(Q\), see A. B. Buan, R. J. Marsh, I. Reiten [Trans. Am. Math. Soc. 359, No. 1, 323-332 (2007; Zbl 1123.16009)].

The article is to be continued in part II [J. Am. Math. Soc. 23, No. 3, 749-790 (2010; Zbl 1208.16017)] on applications of representations of quivers with potentials and their mutations to cluster algebras.

Reviewer: Stanisław Kasjan (Toruń)

##### MSC:

16G20 | Representations of quivers and partially ordered sets |

13F60 | Cluster algebras |

16G10 | Representations of associative Artinian rings |

16S38 | Rings arising from noncommutative algebraic geometry |