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

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.