Gentle algebras arising from surface triangulations

By Ibrahim Assem, Thomas Brustle, Gabrielle Charbonneau-Jodoin, Pierre-Guy Plamondon. 2009

213 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7b4e7fe9dad7f8da32f5f8ed431d0fb6a3766a20

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Citations in the corpus

On the cluster category of a marked surface without punctures (2010, methodology): “In fact, in [22, 2] a quiver with potential (QΓ,WΓ) has been defined for each triangulation Γ of (S,M).”
A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, background): “The Jacobian algebra associated to has been introduced in [7].”
Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions (2012, background): “[15]; and in the unpunctured non-empty boundary case, [3]), and for all tagged triangulations of arbitrarily-punctured surfaces with non-empty boundary (cf.”
A Geometric Model for the Module Category of a Gentle Algebra (2018, methodology): “[22]), and as Jacobian algebras associated to triangulations of unpunctured surfaces with marked points on the boundary [2, 31].”
Representations of Algebras and Related Topics (2011, background): “odJac(Qτ,Wτ) , where FT τ [−1] = HomC(Tτ[−1],−) ≃HomC(Tτ,−[1]). Remark 3.15. In the case of an unpunctured surface (Σ,M), Bru¨stle and Zhang study the category C(Σ,M) very precisely in [21] (see also [10]), and show an analogue of Theorem 2.7, that is any cluster-tilting object of C(Σ,M) is in IY(Σ,”
The representation type of Jacobian algebras (2013, background): “unctured (i.e. all marked points are in the boundary of Σ), then the Jacobian algebra P(Q(τ),S(τ)) is a gentle algebra in the sense of Assem and Skowron´ski [AS] for all triangulations τof (Σ,M), see [ABCP]. (2) If the boundary of Σ is non-empty, and (Σ,M) is not a monogon, and not a torus with |M|=”
Trivial extensions of gentle algebras and Brauer graph algebras (2014, methodology): “In [1] and in [16], a finite dimensional algebra, the so-called Jacobian algebra, is associated to the triple (S,M, T ).”
Cluster categories for marked surfaces: punctured case (2013, background): “In the unpunctured case, – Assem-Brüstle-Charbonneau Jodoin-Plamondon(=ABCP) [3] proved that the Jacobian algebra of such a quiver with potential is a gentle algebra.”