Classification of modules for infinite-dimensional string algebras

TL;DR

This paper extends the classification of modules over string algebras to include certain infinite-dimensional cases, using a functorial filtration approach to classify modules via string and band modules.

Contribution

It introduces a framework for classifying modules over infinite-dimensional string algebras, broadening the scope beyond finite-dimensional cases.

Findings

Finitely generated and artinian modules are classified by string and band modules.

The classification includes infinite words parameterizing string modules.

The approach generalizes known classifications for finite-dimensional modules.

Abstract

We relax the definition of a string algebra to also include infinite-dimensional algebras such as k[x,y]/(xy). Using the functorial filtration method, which goes back to Gelfand and Ponomarev, we show that finitely generated and artinian modules (and more generally finitely controlled and pointwise artinian modules) are classified in terms of string and band modules. This subsumes the known classifications of finite-dimensional modules for string algebras and of finitely generated modules for k[x,y]/(xy). Unlike in the finite-dimensional case, the words parameterizing string modules may be infinite.