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2008

Quivers with potentials associated to triangulated surfaces

Labardini-Fragoso, Daniel

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We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky.

To each ideal triangulation of a bordered surface with marked points we associate a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc.
We prove that if the surface has non-empty boundary, then the quivers with potentials associated to its triangulations are rigid and hence non-degenerate.

Quivers with potentials associated to triangulated surfaces · Around

Built on

S. Fomin and A. Zelevinsky. Cluster algebras I: Foundations
2002
Earlier work this paper cites.
S. Fomin and A. Zelevinsky. Cluster algebras II: Finite type classification
2003
Earlier work this paper cites.
M. Gekhtman, M. Shapiro and A. Vainshtein. Cluster algebras and Weil-Petersson forms
2005
Earlier work this paper cites.
P. Caldero, F. Chapoton and R. Schiffler. Quivers with relations arising from clusters (An case)
2006
Earlier work this paper cites.

Similar

T. Brüstle. Gentle algebras given by surface triangulations
2007
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V. V. Fock and A. B. Goncharov. Dual Teichmüller and lamitation spaces
2007
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R. Schiffler. Geometric realizations of cluster categories
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I. Assem, T. Brüstle, G. Charbonneau-Jodoin and P-G. Plamondon. Gentle algebras arising from surface triangulations
Cited in the paper.
H. Derksen, J. Weyman and A. Zelevinsky. Quivers with potentials and their representations I: Mutations
Original
Cited in the paper.
S. Fomin, M. Shapiro and D. Thurston. Cluster algebras and triangulated surfaces, part I: Cluster complexes
Cited in the paper.
S. Fomin and D. Thurston. Cluster algebras and triangulated surfaces, part II: Lambda lengths
Cited in the paper.
D. Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces, part II: Arc representations
Cited in the paper.

Then

A. Zelevinsky. Mutations for quivers with potentials: Oberwolfach talk, april 2007
Original
2007
Later among the works it cites.
P-G. Plamondon. Algèbres inclinées amassées aimables
2008
Closest in time.
R. Schiffler. A geometric model for cluster categories of type D n D_{n}
2008
Closest in time.

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