A Geometric Model for the Module Category of a Gentle Algebra

Baur, Karin; Coelho, Simoes, Raquel

doi:10.1093/imrn/rnz150

Cited by 35 publications

(59 citation statements)

References 62 publications

(107 reference statements)

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“…The following definition generalises the one of [MP17] for the case whereS is a disk. A very similar definition appears in [BCS18] in a slightly different context under the name "permissible arc". Definition 3.8.…”

Section: Accordion Complex Slalom Complex and Non-crossing Complexmentioning

confidence: 94%

“…In most of the above cases, the algebras obtained are gentle algebras. It has also been shown in [BCS18] that any gentle algebra is obtained from a dissection of a surface, and that the module category of the algebra can be interpreted by using curves on the surface. In the case where the surface is a polygon, the τ -tilting theory of the algebra of a dissection has been studied in [PPP17,PPS18].…”

Section: Introductionmentioning

confidence: 99%

“…Our construction has the advantage that it easily yields the two dual dissections of the surface at the same time (the dissection and dual lamination of [OPS18]). Note that our dissections are always cellular, while those in [BCS18] can be arbitrary. Remarkably, gentle algebras and surfaces were linked recently in [HKK17,LP18], where the Fukaya category of the surface is shown to be equivalent to the bounded derived category of the associated gentle algebra.…”

Section: Introductionmentioning

confidence: 99%

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Non-kissing and non-crossing complexes for locally gentle algebras

2019

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Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to τ -tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.YP, VP and PGP were partially supported by the French ANR grant SC3A (15 CE40 0004 01). VP was partially supported by the French ANR grant CAPPS (17 CE40 0018). 1 arXiv:1807.04730v2 [math.CO] 13 Jul 20182.1. Locally gentle bound quivers and their blossoming quivers. We consider a bound quiverQ = (Q, I), formed by a finite quiver Q and an ideal I of the path algebra kQ (the k-vector space generated by all paths in Q, including vertices as paths of length zero, with multiplication induced by concatenation of paths) such that I is generated by linear combinations of paths of length at least two. Note that we do not require that the quotient algebra kQ/I be finite dimensional. The following definition is adapted from [BR87].Definition 2.1. A locally gentle bound quiverQ := (Q, I) is a (finite) bound quiver where (i) each vertex a ∈ Q 0 has at most two incoming and two outgoing arrows, (ii) the ideal I is generated by paths of length exactly two, (iii) for any arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that t(α) = s(β) and αβ / ∈ I (resp. αβ ∈ I) and at most one arrow γ ∈ Q 1 such that t(β) = s(γ) and βγ / ∈ I (resp. βγ ∈ I). The algebra kQ/I is called a locally gentle algebra. A gentle bound quiver is a locally gentle bound quiverQ such that the algebra kQ/I is finite-dimensional; the algebra is then called a gentle algebra.Definition 2.2. A locally gentle bound quiverQ is complete if any vertex a ∈ Q 0 is incident to either one (a is a leaf) or four arrows (a is an internal vertex). The pruned subquiver of a

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Section: Accordion Complex Slalom Complex and Non-crossing Complexmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-kissing and non-crossing complexes for locally gentle algebras

2019

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“…T (M(v), M(u)) = 0 if and only if u = u (1) ← w → u (2) and v = v (1) → w ← v (2) . Additionally, in this case, (1) → w → u (2)…”

Section: Is a String In T In This Case ξ Is The Unique Nonsplit Exmentioning

confidence: 99%

Oriented Flip Graphs, Noncrossing Tree Partitions, and Representation Theory of Tiling Algebras

Garver¹,

McConville²

2019

Glasgow Math. J.

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The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

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“…They were introduced in [AS87] in the study of iterated tilted algebras of type A m , but have recently appeared in connection with dimer models [Boc12,Bro12], enveloping algebras of some Lie algebras [HK06], cluster algebras and categories arising from triangulated surfaces [LF09,ABCJP10], m-Calabi-Yau tilted algebras [GE17,GE18], non-kissing complexes of grids and associated objects [McC17, GM18, PPP17, BDM + 17], non-commutative nodal curves [BD18], and partially wrapped Fukaya categories [HKK17,LP18]. Surface models have been introduced to study the category representations of a gentle algebra and associated categories [BCS18,OPS18,PPP18].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%