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      When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I

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          Abstract

          For a hereditary, finite dimensional k-algebra A the Coxeter transformation is a linear endomorphism ΦA:K0(modA)K0(modA) of the Grothendieck group K0(modA) of modA such that ΦA[M]=[τAM] for every non-projective indecomposable A-module M, that is the Auslander-Reiten translation τA induces a linear endomorphism of K0(modA). It is natural to ask whether there are other algebras A having a linear endomorphism ΦAEndZ(K0(modA)) with ΦA[M]=[τAM] for all non-projective MindA. We will show that this is the case for all Nakayama algebras. Conversely, we will show that if an algebra A=kQ/I, where Q is a non-acyclic quiver and IkQ is an admissible ideal, admits such a linear endomorphism then A is already a cyclic Nakayama algebra.

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          Author and article information

          Journal
          11 March 2022
          Article
          2203.06238
          1f665bfa-6464-4289-b619-67d55ce01be9

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          16G70, 16E20
          math.RT

          Algebra
          Algebra

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