Generalized bound path algebras and their representations

Author(s)

Viktor Chust Bugno Pires de Almeida

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2020-12-14

Examining board members

Flavio Ulhoa Coelho; Edson Ribeiro Alvares; Sonia Elizabeth Trepode

• Advisor: Flavio Ulhoa Coelho

Abstract

The concept of generalized path algebras (here abbreviated by GPA), in the way it is treated here, was introduced by F. U. Coelho and S. X. Liu in (Coelho, Liu, 2000). The aim of this dissertation is to give a deeper knowledge about these algebras and their representations, listing not only results already existent in the literature but also new approaches which are to be introduced here. Let $\\Gamma$ be a quiver, and let $\\calA = \\{A_i: i \\in \\Gamma_0 \\}$ be a family of algebras, where $\\Gamma_0$ is the set of vertices of $\\Gamma$. A generalized path algebra $k(\\Gamma,\\calA)$ is defined as the vector space having as its basis the set of paths over $\\Gamma$ interposed by elements from the $A_i$ which correspond to each vertex. Multiplication in $k(\\Gamma,\\calA)$ is subsequently defined by juxtaposition of paths and using the internal multiplications of the algebras $A_i$. Another work which will be fundamental here is the article (Ibáñez Cobos, Navarro, López Peña, 2008). There, the authors R. M. Ibáñez-Cobos, G. Navarro and J. López-Peña obtain generalizations of two well-known theorems due to P. Gabriel, which originally deal with ordinary path algebras (see (Auslander, Reiten, Smalo, 1995), (Assem, Coelho, 2020), e.g., for an introduction to these theorems). One of the problems we deal with is to decide whether a given algebra is isomorphic to a GPA in a non-trivial manner. The approach to this problem is even more interesting when we allow the GPA\'s to have relations. By adapting the definitions and results from (Ibáñez Cobos, Navarro, López Peña, 2008) to this new context, it is possible to study this problem using criteria which have combinatorial type. Given a GPA $k(\\Gamma,\\calA)$, we say that a property of algebras or representations holds locally if it holds for every algebra belonging to the family $\\calA$, and we say it holds globally if it holds for the algebra $k(\\Gamma,\\calA)$. The relationship between local and global properties is another relevant problem to be discussed here. In the literature, examples of these properties appear in (Külshammer, 2017), (Wang, 2006). Still in this context, by a deeper insight into a discussion present in (Li, Ye, 2015), it is possible to describe the representations that correspond to simple, projective and injective modules. (AU)

• FAPESP's process: 18/18123-5 - Covering techniques in the study of degrees of irreducible morphisms
• Grantee: Viktor Chust Bugno Pires de Almeida
• Support Opportunities: Scholarships in Brazil - Master