Auslander-Reiten components for clans
It is proved that if the repetitive algebra \(\widehat\Lambda\) of a gentle algebra \(\Lambda\) is of non-polynomial growth, then the stable Auslander-Reiten quiver of \(\widehat\Lambda\) contains a finite number of components of type \(\mathbb{Z}\mathbb{A}_\infty\). Then, if the gentle algebra \(\Lambda\) has finite global dimension, \(\Lambda\) is derived equivalent to a hereditary algebra of type \(\mathbb{A}\) or \(\widetilde\mathbb{A}\), if and only if the stable Auslander-Reiten quiver of \(\widehat\Lambda\) does not contain components of type \(\mathbb{Z}\mathbb{A}_\infty\) or \(\mathbb{Z}\mathbb{A}_\infty^\infty\). These results extend to a more general situation, that is, when \(\widehat\Lambda\) is quasi-clanish. The notion of a quasi-clanish algebra generalizes the notion of clanish algebras and of special biserial algebras. Also, they study certain skew group algebras. All these results are independent of the base field \(k\).
scientific article; zbMATH DE number 1444141