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Representation theory of partial relation extensions

Volume 155 / 2019

Ibrahim Assem, Juan Carlos Bustamante, Julie Dionne, Patrick Le Meur, David Smith Colloquium Mathematicum 155 (2019), 157-186 MSC: Primary 16G10; Secondary 16G70, 16E30. DOI: 10.4064/cm7511-3-2018 Published online: 8 November 2018


Let $C$ be a finite dimensional algebra of global dimension at most two. A partial relation extension is any trivial extension of $C$ by a direct summand of its relation $C$-$C$-bimodule. When $C$ is a tilted algebra, this construction provides an intermediate class of algebras between tilted and cluster tilted algebras. The text investigates the representation theory of partial relation extensions. When $C$ is tilted, any complete slice in the Auslander–Reiten quiver of $C$ embeds as a local slice in the Auslander–Reiten quiver of the partial relation extension. Moreover, a systematic way of producing partial relation extensions is introduced by considering direct sum decompositions of the potential arising from a minimal system of relations of $C$.


  • Ibrahim AssemDépartement de Mathématiques
    Université de Sherbrooke
    Sherbrooke, Québec, Canada J1K 2R1
  • Juan Carlos BustamanteDépartement de Mathématiques
    Université de Sherbrooke
    Sherbrooke, Québec, Canada J1K 2R1
    Current address:
    Mathematics Department
    Champlain College – Lennoxville
    2580 Rue College
    Sherbrooke, Québec, Canada J1M 2K3
  • Julie Dionne
  • Patrick Le MeurLaboratoire de Mathématiques
    Université Blaise Pascal \& CNRS
    Complexe Scientifique Les Cézeaux
    BP 80026
    63171 Aubière Cedex, France
    Current adress:
    Université Paris Diderot, Sorbonne Université
    CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche
    F-75013 Paris, France
  • David SmithDepartment of Mathematics
    Bishop’s University
    Sherbrooke, Québec, Canada J1M 1Z7

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