Clannish algebras are of amenable representation type

TL;DR

This paper proves that clannish algebras have an amenable representation type by connecting their module structure to graph planarity, extending previous results on string algebras.

Contribution

It provides a new proof that clannish algebras are of amenable type, utilizing graph-theoretic methods related to the planarity of coefficient quivers.

Findings

String algebras are amenable, proven via planarity of their coefficient quivers.

Clannish algebras are also of amenable type, extending previous results.

Graph theory techniques are effective in studying algebra representation types.

Abstract

We revisit G. Elek's notion of amenable representation type, where algebras are characterised by every indecomposable module being "almost" the direct sum of modules of bounded dimension. We give a new proof of his result that string algebras are amenable that relies on the planarity of the coefficient quivers of indecomposable string and band modules. Using this connection to graph theory, we then show that clannish algebras are also of amenable type.