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Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

Part of: Rings and algebras arising under various constructions Arithmetic rings and other special rings

Published online by Cambridge University Press:  01 November 2012

Giovanni Cerulli Irelli  and
Daniel Labardini-Fragoso
Giovanni Cerulli Irelli
Affiliation:
Mathematisches Institut, Universität Bonn, Bonn, 53115, Germany (email: cerulli@math.uni-bonn.de)
Daniel Labardini-Fragoso
Affiliation:
Mathematisches Institut, Universität Bonn, Bonn, 53115, Germany (email: labardini@math.uni-bonn.de)
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Abstract

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To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

Keywords

quiver with potentialJacobian algebratagged triangulationcluster algebracluster monomial

MSC classification

Secondary: 16S99: None of the above, but in this section 13F60: Cluster algebras
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1833 - 1866
Copyright
Copyright © Foundation Compositio Mathematica 2012

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