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

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arxiv 1904.02555 v3 pith:WOTHNLQF submitted 2019-04-04 math.RT math.SG

Claire Amiot , Pierre-Guy Plamondon , Sibylle Schroll This is my paper · ORCID
classification math.RTmath.SG
keywords gentlealgebrasadmissiblederiveddissectionsinvariantsurfacealgebra
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abstract

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 $\mathbb Z_2$, we translate this to a numerical complete derived invariant for gentle algebras.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Resolving subcategories for gentle algebras III: Tilting modules for gentle tree algebras

    math.RT 2026-06 unverdicted novelty 6.0 of 10

    Gives a combinatorial realization of the Auslander-Reiten correspondence between resolving subcategories and tilting modules for gentle tree algebras.

  2. Tilting-completion for gentle algebras

    math.RT 2024-12 conditional novelty 6.0 of 10

    For gentle algebras, almost-tilting modules always complete to tilting modules with at most 2n complements, matching a modified version of Happel's conjecture.

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