Domestic trivial extensions of simply connected algebras

Abstract

Let $A$ be a finite-dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field. It is called simply connected it it is triangular and, for any presentation of $A$ as a bound quiver algebra, the fundamental group of its bound quiver is trivial. Let $T(A)$ denote the trivial extension of $A$ by its minimal injective cogenerator. We show that, if $A$ is simply connected, then the following conditions are equivalent: (i) $T(A)$ is representation-infinite and domestic, (ii) $T(A)$ is 2-parametric, (iii) there exists a representation-infinite tilted algebra $B$ of Euclidean type $\tilde{D}_{n}$ or $\tilde{E}_{p}$ such that $T(A)_{\rightarrow}\sim T(B)$ , (iv) $A$ is an iterated tilted algebra of type $\tilde{D}_{n}$ or $\tilde{E}_{p}$ .