Representações de álgebras associativas e suas álgebras repetitivas

Abstract

In 1983, Hughes and Waschbüsch introduced the concept of repetitive algebras in [HW83] to study the structure of trivial extensions of tilted algebras, exploring their properties. They characterized these extensions, particularly in relation to representation theory, demonstrating how these constructions can enhance the understanding of modules and morphisms in tilted-type algebras. In 2018, Hernán Giraldo studied irreducible morphisms and almost split sequences in [GIR18], leveraging the interpretation of the module category over repetitive algebras in terms of certain types of cochain complexes. In this work, we will study Gabriel's Theorem and the relationship between quiver representations and modules over path algebras, exploring the category isomorphism with an emphasis on simple, projective, and injective modules, based on the approaches presented in [Gir15] and [ASS06]. We will employ J. Schröer's techniques to explicitly describe repetitive quivers (see [Schr99]) and establish a relationship, via category isomorphism, between the modules over the path algebra of these quivers and cochain complexes tensorized by the corresponding algebra (refer to [BR24] and [GIR18] for further details).