Connected gradings and fundamental group

TL;DR

This paper develops a method to compute the fundamental group of algebras by analyzing connected gradings and their relation to Galois coverings, providing explicit calculations for various cases.

Contribution

It introduces a framework linking connected gradings and Galois coverings to facilitate the explicit computation of algebraic fundamental groups.

Findings

Explicit computations of fundamental groups for several algebras

Establishment of a relation between gradings and Galois coverings

Development of a method for using gradings to determine fundamental groups

Abstract

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of $A$, which allows its computation using complete lists of gradings.