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Quivers with potentials and their representations. I: Mutations. (English) Zbl 1204.16008
Mutations 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.

16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
16G10 Representations of associative Artinian rings
16S38 Rings arising from noncommutative algebraic geometry
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