Tame hereditary path algebras and amenability

TL;DR

This paper investigates the amenable representation type of tame hereditary path algebras, proving they are of this type, and shows that wild acyclic quivers over finite fields are not amenable, extending previous conjectures.

Contribution

It proves tame hereditary path algebras of extended Dynkin quivers are of amenable type, broadening the class of known amenable algebras without relying on string algebra results.

Findings

Tame hereditary path algebras of extended Dynkin type are of amenable type.

Path algebras of wild acyclic quivers over finite fields are not amenable.

Extends the conjecture on amenability to a new class of tame algebras.

Abstract

In this note we are concerned with the notion of amenable representation type as defined in a recent paper by G\'abor Elek. Roughly speaking, an algebra is of amenable type if for all $\varepsilon > 0$, every finite-dimensional module has a submodule which is a direct sum of modules which are small with respect to $\varepsilon$ such that the quotient is also small in that respect. We will show that the tame hereditary path algebras of quivers of extended Dynkin type over any field $k$ are of amenable type, thus extending a conjecture in the aforementioned paper to another class of tame algebras. In doing so, we avoid using already known results for string algebras. We also show that path algebras of wild acyclic quivers over finite fields are not amenable.