Classification results for \(n\)-hereditary monomial algebras. (English) Zbl 07808733
Auslander-Reiten theory has proven to be a central tool in the study of the representation theory of Artin algebras. M. H. Sandøy and L.-P. Thibault classify \(n\)-hereditary monomial algebras in three natural contexts. They give a classification of the \(n\)-hereditary truncated path algebras by showing that they are exactly the \(n\)-representation-finite Nakayama algebras. Next, the authors classify partially the \(n\)-hereditary quadratic monomial algebras. In the case \(n = 2\), they prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating \(k\mathbb{A}_3 \otimes_k k\mathbb{A}_3\), where \(\mathbb{A}_3\) is the Dynkin quiver of type \(A\) with bipartite orientation. In the case \(n \geq 3\), they show that the only \(n\)-representation finite algebras are the \(n\)-representation-finite Nakayama algebras with quadratic relations.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
Keywords:
\(n\)-hereditary algebra; \(n\)-representation-finite algebra; preprojective algebra; selfinjective algebra; Calabi-Yau algebra; Auslander-Reiten theoryReferences:
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