Substructures of algebras with weakly non-negative tits form

Abstract

Let 𝐴 = kQ/I be a finite dimensional basic algebra over an algebraically closed field k presented by its quiver Q with relations I. A fundamental problem in the representation theory of algebras is to decide whether or not 𝐴 is of tame or wild type. In this paper we consider triangular algebras 𝐴 whose quiver Q has no oriented paths. We say that A is essentially sincere if there is an indecomposable (finite dimensional) A-module whose support contains all extreme vertices of Q We prove that if A is an essentially sincere strongly simply connected algebra with weakly non-negative Tits form and not accepting a convex subcategory which is either representation-infinite tilted algebra of type Ẽₚ or a tubular algebra, then 𝐴 is of polynomial growth (hence of tame type).