Tame systems of linear and semilinear mappings

TL;DR

This paper extends classical theorems on quivers to systems of linear and semilinear mappings represented by directed graphs, classifying their types and properties.

Contribution

It generalizes Gabriel's and Nazarova et al.'s theorems to include semilinear mappings in graph representations.

Findings

Classifies systems of linear and semilinear mappings as finite or tame types.

Extends Gabriel's theorems to new classes of representations.

Provides a framework for understanding complex graph-based mappings.

Abstract

We study systems of linear and semilinear mappings considering them as representations of a directed graph $G$ with full and dashed arrows: a representation of $G$ is given by assigning to each vertex a complex vector space, to each full arrow a linear mapping, and to each dashed arrow a semilinear mapping of the corresponding vector spaces. We extend to such representations the classical theorems by Gabriel about quivers of finite type and by Nazarova, Donovan, and Freislich about quivers of tame types.