The cluster category of a surface with punctures via group actions

2017/07/06 by Claire Amiot, Amiot, Claire, Plamondon, Pierre-Guy · 2 citations
Mathematics · #16G20 #16G70 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1707.01834

openalex publication_date 2017/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient by this action recovers the original surface. We show that the group acts on the quivers with potentials associated to the surfaces, and that their Ginzburg dg algebras are skew group algebras of each other, up to Morita equivalence. We then use these results to construct functors between the generalized cluster categories associated to the triangulations. This allows us to give a complete description of the indecomposable objects of these categories in terms of curves on the surface, when the surface has punctures and non-empty boundary.

Cited by

• Induction in perverse schobers and cluster tilting theory
• Quivers with potentials and actions of finite abelian groups

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