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On Auslander-Reiten components of string and band complexes for a class of symmetric special biserial algebra

Research paper by Hernán A. Giraldo, Ricardo Rueda-Robayo, José A. Vélez-Marulanda

Indexed on: 06 Apr '20Published on: 02 Apr '20Published in: arXiv - Mathematics - Representation Theory



Abstract

Let $\mathbf{k}$ be an algebraically closed field. In this article we introduced the class of special biserial $\mathbf{k}$-algebras that are of {\it Boundarenko's type}, which contains that of gentle algebras. In particular, if $\Lambda$ is a $\mathbf{k}$-algebra in this class, then string and band complexes (in the sense of V. Bekkert and H. Merklen) are definable over $\Lambda$, which under certain conditions they are indecomposable objects in the category $\mathcal{K}^b(\text{proj-}\Lambda)$ of perfect complexes over $\Lambda$. We prove that if $\Lambda$ is a symmetric special biserial $\mathbf{k}$-algebra of Boudarenko's type, $P^\bullet$ is either a string or a band complex over $\Lambda$ which is also indecomposable in $\mathcal{K}^b(\text{proj-}\Lambda)$, and $\mathfrak{C}$ is the component of the Auslander-Reiten quiver of $\mathcal{K}^b(\text{proj-}\Lambda)$ containing $P^\bullet$, then $P^\bullet$ lies in the rim of $\mathfrak{C}$. In this situation, we also give a full description of the representatives of the Auslander-Reiten translation orbits in $\mathfrak{C}$