A representation embedding for algebras of infinite type

TL;DR

This paper constructs a representation embedding from a principal ideal domain to any infinite-type algebra, and applies it to prove a variation of the Brauer-Trall Conjecture, revealing infinite families of indecomposables.

Contribution

It introduces a new representation embedding for infinite-type algebras and proves a related variation of the Brauer-Trall Conjecture.

Findings

Existence of a bounded principal ideal domain embedding for infinite-type algebras

Infinite families of non-isomorphic indecomposables with fixed endolength

Extension of the Brauer-Trall Conjecture to infinite representation type

Abstract

We show that for any finite-dimensional algebra $\Lambda$ of infinite representation type, over a perfect field, there is a bounded principal ideal domain $\Gamma$ and a representation embedding from $\Gamma -$mod into $\Lambda -$mod. As an application, we prove a variation of the Brauer-Trall Conjecture II: finite-dimensional algebras of infinite-representation type admit infinite families of non-isomorphic finite-dimensional indecomposables with fixed endolength, for infinitely many endolengths.