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Proceedings of the London Mathematical Society Advance Access originally published online on November 14, 2008
Proceedings of the London Mathematical Society 2009 98(3):797-839; doi:10.1112/plms/pdn051
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© 2008 London Mathematical Society

Quivers with potentials associated to triangulated surfaces

Daniel Labardini-Fragoso

Department of Mathematics
Northeastern University
Boston, MA 02115
USA

Received 18 April 2008. Revision received 12 August 2008.

We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin–Shapiro–Thurston, and quivers with potentials (QPs) and their mutations introduced by Derksen–Weyman–Zelevinsky. To each ideal triangulation of a bordered surface with marked points, we associate a QP, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective QPs are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the QPs associated to its triangulations are rigid and hence non-degenerate.

2000 Mathematics Subject Classification 16G99, 16S99, 57N05, 57M50.

This work was partially supported by Prof. Andrei Zelevinsky's NSF grant and Prof. José Antonio de la Peña's SNI grant.


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