On the stable radical of some non-domestic string algebras

TL;DR

This paper introduces new concepts and techniques to analyze the stable radical of non-domestic string algebras, focusing on bridge quivers and classifying certain algebra types based on graph map properties.

Contribution

It defines prime bands and finite bridge quivers, and develops recursive systems to study stable radicals, classifying meta-$igcup$-cyclic and meta-torsion-free string algebras.

Findings

Meta-$igcup$-cyclic algebras characterized by string substring property.

Meta-torsion-free algebras exhibit a dichotomy in graph map ranks.

Stable ranks are limited to , +1, +2.

Abstract

We introduce the concept of a prime band in a string algebra $\Lambda$ and use it to associate to $\Lambda$ its finite bridge quiver. Then we introduce a new technique of `recursive systems' for showing that a graph map between finite dimensional string modules lies in its stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have `meta-torsion-free' string algebras that are characterized by a dichotomy statement for ranks of graph maps between string modules--such maps either have finite rank or are in the stable radical. Their stable ranks can only take values from $\{\omega,\omega+1,\omega+2\}$.