Functorial methods in representation theory with applications to monomial algebras

• Autor(a) principal: Lobo, Daniel Negreiros
• Data de Publicação: 2024
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-05082024-144708/
• topic: Álgebras monomiais; Álgebras string; Covering theory; Monomial algebras; Representation theory of associative algebras; String algebras; Teoria das representações de álgebras associativas; Teoria de recobrimento

String algebras have become a staple of modern research into the representation theory of finite dimensional associative algebras. The indecomposable modules for these algebras have been known since Butler and Ringel introduced them, and they come in two flavours: the string and band modules. The goal of this thesis is to develop categorical ideas to motivate these classes of modules and the tools used to work with them. Explicitly, we will look to covering theory in order to define string and band modules. We restrict ourselves to the locally bounded case for the most part, as it is exactly what is needed for our purposes. This is not only an excuse to go through a pleasant stroll through the theory of Galois covers, but also a valuable insight which lead to the classification of morphisms between string modules by Crawley-Boevey and later extended to morphisms between band modules by Henning Krause. We will also study the functor category of a Krull-Schmidt category closely. This will be done in order to develop the functorial filtration method, which was a key part in the classification theorem of indecomposable modules for string algebras. We close off the text by providing an overview of how the method of functorial filtration was used in this particular case.