When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866914561640103936 |
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| author | Klapproth, Carlo |
| author_facet | Klapproth, Carlo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_06238 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I Klapproth, Carlo Representation Theory 16G70, 16E20 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. |
| title | When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I |
| topic | Representation Theory 16G70, 16E20 |
| url | https://arxiv.org/abs/2203.06238 |