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Selfinjective quivers with potential and 2-representation-finite algebras

Part of: Representation theory of rings and algebras Homological algebra

Published online by Cambridge University Press:  28 September 2011

Martin Herschend  and
Osamu Iyama
Martin Herschend
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan (email: martin.herschend@gmail.com)
Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan (email: iyama@math.nagoya-u.ac.jp)
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Abstract

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We study quivers with potential (QPs) whose Jacobian algebras are finite-dimensional selfinjective. They are an analogue of the ‘good QPs’ studied by Bocklandt whose Jacobian algebras are 3-Calabi–Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under iterated mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.

Keywords

2-representation-finite algebraquiver with potential3-preprojective algebraselfinjective algebracluster tiltingmutation

MSC classification

Secondary: 16G10: Representations of Artinian rings 18G20: Homological dimension
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1885 - 1920
Copyright
Copyright © Foundation Compositio Mathematica 2011

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