From finite to infinite length modules over tame hereditary algebras

TL;DR

This paper provides a comprehensive introduction to infinite dimensional representations over tame hereditary algebras, detailing pure-injective modules and highlighting torsionfree divisible modules as direct sums of the generic module.

Contribution

It offers a complete description of pure-injective modules over tame hereditary algebras, emphasizing the role of the generic module in torsionfree divisible modules.

Findings

Complete classification of pure-injective modules.

Identification of torsionfree divisible modules as sums of the generic module.

Provides foundational understanding for infinite dimensional representations.

Abstract

A self-contained introduction to infinite dimensional representations over a tame hereditary algebra is provided, assuming a basic knowledge of the category of finite dimensional representations. This includes a complete description of all pure-injective modules. Of particular interest are the torsionfree divisible modules, which are precisely the direct sums of copies of the unique generic module.