Selfinjective algebras with hereditary stable slice (Q2417795)

Let \(A\) be a finite-dimensional associative self-injective algebra over a field \(K\). This paper considers algebras \(A\) which admit a \emph{finite stable slice}; that is, algebras for which there exists a finite collection of indecomposable finitely generated \(A\)-modules (subject to some homological conditions) from which allow one to recover the Auslander-Reiten (AR) quiver of \(A\) in a precise way. These collections are identified with full (valued) subquivers of the AR quiver. More specifically, the authors consider finite stable slices \(\Delta\) which are \emph{hereditary} (meaning the endomorphism ring of the direct sum of the corresponding modules is hereditary with AR quiver \(\Delta^{\mathrm{op}}\)) and either \emph{almost right regular} (meaning the radical of a projective can only lie in \(\Delta\) if it is a sink of \(\Delta\)) or almost left regular (meaning that if \(P\) is indecomposable projective (=injective) and \(P/\mathrm{soc}P \in \Delta\), then \(P/\mathrm{soc}P\) is a source of \(\Delta\)). The main result shows that such a finite stable slice exists if and only if there exists an algebra \(A' = \overline{B}/(\varphi\nu_{\widehat{B}})\) with \(A/\mathrm{soc}A \cong A'/\mathrm{soc}A'\) such that \(B\) is a tilted algebra with repetitive category \(\widehat{B}\), \(\varphi\) is a positive automorphism of \(\widehat{B}\), and \(\nu_{\widehat{B}}\) is the Nakayama functor in \(\widehat{B}\).