Auslander-Reiten quivers of local orders of finite lattice type
Let R be a complete d.v.r. with quotient field K and \(\Lambda\) a representation-finite local R-order in a semisimple K-algebra A. The classification of Auslander-Reiten quivers \({\mathcal A}(\Lambda)\) of such \(\Lambda\) is completed in this paper. Previously, the author classified \({\mathcal A}(\Lambda)\) for \(\Lambda\) Gorenstein, while \textit{K. W. Roggenkamp} [J. Algebra 85, 449-476 (1983; Zbl 0526.16007)] classified \({\mathcal A}(\Lambda)\) for Bäckström orders \(\Lambda\). Using \textit{Yu. A. Drozd} and \textit{V. V. Kirichenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 715-736 (1973; Zbl 0291.16005)], it turns out that there are essentially two new translation quivers \({\mathcal A}(\Lambda)\) which do not stem from Gorenstein or Bäckström orders \(\Lambda\). Namely, if \(\Lambda\) is non-Gorenstein and the multiplier \(O_{\ell}(Rad \Lambda)\) of the Jacobson radical of \(\Lambda\) is not a Bass order, then one of the two special cases arises. The corresponding two quivers \({\mathcal A}(\Lambda)\) are given explicitly.
scientific article; zbMATH DE number 4008590