Prime ideals in categories of representations of quivers of type $A$

TL;DR

This paper classifies prime tensor ideals in categories of quiver representations, specifically for zigzag quivers with integer vertices, revealing a bijection with prime ideals of a Boolean algebra.

Contribution

It provides a complete classification of prime tensor ideals in categories of zigzag quiver representations, linking them to Boolean algebra prime ideals.

Findings

Prime tensor ideals correspond to prime ideals of a Boolean algebra.

Classification applies to zigzag quivers with bounded path length.

Establishes a canonical bijection between these ideals.

Abstract

We study prime tensor ideals in tensor abelian categories of quiver representations. Specifically, we classify the prime tensor ideals in the category of representations of zigzag quivers (with bounded path length) whose vertex set is the set of integers. We show that prime tensor ideals in these categories are in canonical bijection with prime ideals of a Boolean algebra, the power set of integers.