Prime spectra of derived quiver representations

TL;DR

This paper investigates the prime spectra of derived categories of quiver representations, revealing limitations in their ability to recover the original quiver, and introduces an algebraic method to recover the path algebra from tensor categories.

Contribution

It demonstrates that Balmer's prime spectrum does not fully recover the quiver and proposes an algebraic construction to recover the path algebra from tensor categories.

Findings

Balmer's spectrum fails to recover the quiver

An algebra associated to tensor categories can recover the path algebra

Provides new insights into the structure of derived quiver representations

Abstract

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered this way.