Tame tilted algebras with almost regular connecting components (Q2782426)

A component of the AR-quiver of an algebra \(A\) is called almost regular if it has only one non-stable \(\tau_A\)-orbit, which consists of exactly one vertex. The main result of this work is to classify all representation infinite tame tilted algebras with almost regular connecting components, in terms of (families of) quivers and relations. As a consequence, the author can describe the self-injective algebras of Euclidean type which admit almost regular non-periodic components in the Auslander-Reiten quiver. Let \(A\) be a representation infinite tame tilted algebra with almost regular connecting component and let \(X\) be the unique projective injective \(A\)-module. Then there exists \(B_1\) and \(B_2\) tilted of Euclidean type such that \(A=B_1[\text{rad }X]=[X/\text{soc }X]B_2\) and \(B\) a product of tilted algebras of Dynkin type, such that \(B[R]=B_1\) and \([R]B=B_2\). The proof is done by considering (using vectorspace categories methods) the possibilities for \(B\) and \(R\) such that \(B[R]\) and \([R]B\) are tilted of Euclidean type.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00038].