Category Equivalences Involving Graded Modules Over Weighted Path Algebras and Weighted Monomial Algebras

TL;DR

This paper establishes an equivalence between categories of graded modules over weighted path algebras and unweighted path algebras, revealing structural similarities and simplifying their analysis.

Contribution

It introduces a method to convert weighted path algebra categories into unweighted ones via a finite graph, enabling easier study of their module categories.

Findings

Category equivalence between weighted and unweighted path algebra modules

Reduction to modules over ultramatricial, von Neumann regular algebras

Simplification of graded module category analysis

Abstract

Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A = kQ/I. Let QGr A denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph Q' with all its arrows placed in degree 1 and an equivalence of categories QGr A = QGr kQ'. A result of Smith now implies that QGr A = Mod S, the category of right modules over an ultramatricial, hence von Neumann regular, algebra S.