Generic representations of wild quivers

TL;DR

This paper demonstrates that generic representations of wild connected directed quivers are unions of finite-length regular subrepresentations, revealing structural properties and limitations of certain module classes.

Contribution

It establishes that generic modules are unions of regular subrepresentations and shows the direct limit closure of the preprojective component lacks generic modules.

Findings

Generic representations are unions of finite-length regular subrepresentations.

The direct limit closure of the preprojective component contains no generic modules.

Abstract

We show that any generic representation of a wild connected directed quiver is the union of its subrepresentations of finite length which are regular. As a consequence, we see that the direct limit closure of the preprojective component does not contain any generic module.