A geometric model for the module category of a gentle algebra

Baur, Karin; Coelho, Simoes, Raquel

doi:10.48550/arxiv.1803.05802

Cited by 6 publications

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“…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: Theorem (410)mentioning

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: Theorem (410)mentioning

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: Theorem (410)mentioning

confidence: 99%

Section: Theorem (410)mentioning

confidence: 99%

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

2019

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“…Recenlty, gentle algebras have been associated to triangulations or dissections of surfaces, in connection with cluster algebras [ABCJP10], with ribbon graphs [Sch15,Sch18] or with Fukaya categories of surfaces [HKK17]. It turns out that any gentle algebra can be obtained from a dissection of a surface; this has led to geometric models for their module categories [BCS18] (building on [CSP16]) and τ -tilting theory [PPP18] (see also [BDM + 17,PPP17]). From the point of view of homological algebra, the class of gentle algebras is of particular interest, since it is closed under derived equivalence [SZ03].…”

Section: Introductionmentioning

confidence: 99%

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Amiot¹,

Plamondon²,

Schroll³

2019

Preprint

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Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over Z 2 , we translate this to a numerical complete derived invariant for gentle algebras.Contents 5 1.3. The line field of an admissible dissection 9 2. Admissible dissections, gentle algebras and derived categories 2.1. The locally gentle algebra of an admissible dissection 2.2. The surface as a model for the derived category 3. Silting objects 4. Derived invariants for gentle algebras 5. Numerical derived invariants via Arf invariants 5.1. Action of the mapping class group 5.2. Application to derived equivalences 6. Reproving known results on gentle algebras 6.1. The class of gentle algebras is stable under derived equivalences 6.2. Gentle algebras are Gorenstein 7. Examples 8. Complete derived invariant for surface cut algebras

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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%