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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.

##### 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
##### Citations:
Zbl 1042.52007; Zbl 1162.16007; Zbl 1123.16009; Zbl 1208.16017
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