Fractional Brauer configuration algebras II: covering theory

TL;DR

This paper develops a covering theory for fractional Brauer configurations, linking it with quiver coverings, and provides explicit constructions and fundamental group calculations.

Contribution

It introduces a universal cover construction, relates fundamental groups of configurations to quiver groups, and establishes an analogy of Van Kampen theorem for these structures.

Findings

Universal cover of fractional Brauer configurations is simply connected.

Fundamental group of a configuration matches that of its quiver with relations.

Coverings induce Galois coverings of associated categories.

Abstract

In this paper, we develop a covering theory for the fractional Brauer configurations and connect it with the coverings of the associated quivers with relations in the sense of Mart\'inez-Villa and de la Pe\~na. Among the results, we show the following: (1) The universal cover of any fractional Brauer configuration is simply connected and we construct explicitly the universal cover of fractional Brauer configurations of type MS; (2) The fundamental group of a fractional Brauer configuration $E$ of type S is isomorphic to the fundamental group of the associated quiver with relations $(Q_E,I_E)$; (3) A (regular) covering of fractional Brauer configurations induces a (Galois) covering of the associated fractional Brauer configuration categories; (4) Set up an analogy of Van Kampen theorem for fractional Brauer configurations and apply it to calculate the fundamental group of any connected Brauer configuration.