Andrei V. Zelevinsky
Jerzy M. Weyman, Gordana G. Todorov, Harm Derksen
Date of Award
Doctor of Philosophy
Department or Academic Unit
College of Arts and Sciences. Department of Mathematics
theoretical mathematics, flip, mutation, potential, quiver, representation, triangulation
We study the behavior of quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky in the combinatorial framework developed by Fomin-Shapiro-Thurston for cluster algebras that arise from bordered Riemann surfaces with marked points.
In Part I we associate to each ideal triangulation of a bordered surface with marked points 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, and have finite-dimensional Jacobian algebra.
In Part II we define, given an arc and an ideal triangulation of a bordered marked surface, a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation of representations.
Labardini-Fragoso, Daniel, "Quivers with potentials associated with triangulations of Riemann surfaces" (2010). Mathematics Dissertations. Paper 15. http://hdl.handle.net/2047/d20000727
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