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

• Claire Amiot ^[1] ; Pierre-Guy Plamondon ^[2] ; Sibylle Schroll ^[3]
  1. [1] Grenoble Alpes University

     Grenoble Alpes University

     Arrondissement de Grenoble, Francia

     
  2. [2] University of Paris-Saclay

     University of Paris-Saclay

     Arrondissement de Palaiseau, Francia

     
  3. [3] University of Cologne

     University of Cologne

     Kreisfreie Stadt Köln, Alemania

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 2, 2023
• Idioma: inglés
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• Resumen

  • 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 Z2, we translate this to a numerical complete derived invariant for gentle algebras.