Gentle algebras arising from surface triangulations
Ibrahim Assem, Thomas Br\"ustle, Gabrielle Charbonneau-Jodoin,, Pierre-Guy Plamondon
TL;DR
This paper constructs gentle, Gorenstein algebras from surface triangulations and characterizes when these algebras are cluster-tilted, linking surface topology to algebraic properties.
Contribution
It introduces a method to associate gentle algebras to surface triangulations and characterizes their cluster-tilted status based on surface topology.
Findings
A(T) is gentle and Gorenstein of dimension one.
A(T) is cluster-tilted if and only if the surface is a disc or annulus.
All cluster-tilted algebras of type A or A tilde are obtained from surface triangulations.
Abstract
In this paper, we associate an algebra A(T) to a triangulation T of a surface S with a set of boundary marking points. This algebra A(T) is gentle and Gorenstein of dimension one. We also prove that A(T) is cluster-tilted if and only if it is cluster-tilted of type A or A tilde, or if and only if the surface S is a disc or an annulus. Moreover all cluster-tilted algebras of type A or A tilde are obtained in this way.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Combinatorial Mathematics · Commutative Algebra and Its Applications