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

TL;DR

This paper investigates when the Auslander-Reiten translation extends linearly on the Grothendieck group, showing it for Nakayama algebras and characterizing algebras with this property.

Contribution

It proves that all Nakayama algebras admit a linear extension of the Auslander-Reiten translation and characterizes non-acyclic connected quiver algebras with this property.

Findings

Nakayama algebras admit linear extensions of the Auslander-Reiten translation.

Non-acyclic connected quiver algebras with such extensions are cyclic Nakayama algebras.

The linear extension property characterizes a specific class of algebras.

Abstract

For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated $A$-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras.