Simply connected tame quasi-tilted algebras
Throughout this review, by algebra is meant a basic, connected, finite dimensional \(K\)-algebra, where \(K\) is an algebraically closed field. Modules will be finitely generated right modules. An algebra \(A\) is called triangular if its ordinary quiver \(Q_A\) has no oriented cycles. \(KQ_A\) will denote the path algebra of \(Q_A\). \(A\) is called simply connected if it is triangular and for each pair \((Q_A,I_\nu)\) we have that the fundamental group \(\pi(Q_A,I_\nu)=1\), where \(I_\nu\) is the kernel of a surjective algebra morphism \(\nu\colon KQ_A\to A\). In this paper the authors conjecture that a quasitilted algebra \(A\) is simply connected if and only if the first Hochschild cohomology group \(H^1(A)=0\). This conjecture is known to be true in the case that \(A\) is a tame tilted algebra and in the present paper the authors show that the conjecture holds also in the case where \(A\) is a tame quasitilted algebra. Other results presented are: (i) A tame quasitilted algebra is strongly simply connected if and only if it is simply connected and \(\mathbb{A}\)-free (it contains no full convex subcategory which is hereditary of type \(\mathbb{A}\)). (ii) The strong simple connectedness of a tame quasitilted algebra is closely related to that of two particular full subcategories \(A^+\) and \(A^-\) [see \textit{I. Assem}, Lect. Notes Pure Appl. Math. 224, 21-29 (2002; Zbl 1004.16011) for definitions]. (iii) As an application the authors study the simple and strong simple connectedness of a semiregular iterated tubular algebra.
scientific article; zbMATH DE number 1787466
scientific article