Quivers with potentials and their representations II: Applications to cluster algebras

Abstract

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors</italic>, and a family of integer polynomials called <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials</italic>. In the case of skew-symmetric exchange matrices we find an interpretation of these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials made in loc. cit.