Quivers with potentials and their representations I: Mutations

By Harm Derksen, Jerzy Weyman, Andrei Zelevinsky. 2007

529 citations recorded by the source corpus.

Abstract.We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi–Yau algebras, cluster algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a52c2db444514d917d6c1829ca93a01dd8908350

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Citations in the corpus

A survey on maximal green sequences (2019, background): “nce and Q1 Ď Q a full subquiver. We wish to show that Q1 has a maximal green sequence as well. We recall the setup of Derksen–Weyman–Zelevinsky’s theory of quivers with potentials and their mutations [35]. Let k be an uncountable ﬁeld. Let kQ be the path algebra and J the twosided ideal of kQ genera”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “ [94]. 7. Categorification via 2-Calabi-Yau categories The extension of the results of the preceding sections to quivers containing oriented cycles is the subject of ongoing research, cf. for example [39] [62] [8]. Here we present an approach based on the fact that many arguments developed for clust”
Reduction of τ-Tilting Modules and Torsion Pairs (2013, background): “Thus C is isomorphic to the Jacobian algebra of the quiver with potential (Q, x1x2x3), see [13] for definitions.”
Dimer models and cluster categories of Grassmannians (2013, methodology): “The deﬁnition of the Jacobian algebra associated to a quiver with potential involves taking a quotient of the completed path algebra (completed with respect to the arrow ideal) by the closure of the ideal generated by the relations determined by the potential; see [ 9 ].”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “e diamond nodes to denote the frozen vertices. 1 2 3 4 5 6 7 Figure1. The ice quiver associated the adaptable word (1321321) and Cartan type A3 We ﬁx the base ﬁeld to be the complex ﬁeld C. Following [DWZ08], choose a generic potential fW for the ice quiver Qe, namely, a generic linear combination o”
Cluster characters for cluster categories with infinite-dimensional morphism spaces (2010, background): “Following [10], we define a quiver with potential as a pair (Q,W ), where Q is a finite quiver and W is an element of the space Pot(Q) = k̂Q/U , where U is the closure of the commutator [k̂Q, k̂Q].”