Galois Coverings of One-Sided Bimodule Problems

TL;DR

This paper uses geometric methods to construct Galois coverings of bimodule problems, providing a framework to classify finite representation type objects through their universal coverings and structural properties.

Contribution

It introduces a geometric approach to analyze bimodule problems, establishing conditions under which they are schurian or contain specific substructures.

Findings

If a bimodule problem is schurian, its universal covering is also schurian.

Problems with certain finiteness conditions either are schurian or contain a dotted loop.

The structure of the basic bigraph determines the problem's classification.

Abstract

Applying geometric methods of $2$-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem $\mathcal A$ is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering $\tilde{\mathcal A}$, either $\mathcal A$ is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.