Rational points of quiver moduli spaces

TL;DR

This paper investigates the structure of rational points on quiver moduli spaces over perfect fields, revealing a decomposition linked to the Brauer group and interpreting it through twisted quiver representations.

Contribution

It introduces a new decomposition of the rational points of quiver moduli spaces indexed by the Brauer group, using twisted quiver representations for modular interpretation.

Findings

Decomposition of fixed locus indexed by Brauer group elements

Modular interpretation via quiver representations over division algebras

Different forms of moduli spaces from twisted quiver representations

Abstract

For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\bar{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. We provide a modular interpretation of this decomposition using quiver representations over division algebras, and we reinterpret this description using twisted quiver representations. We also see that moduli spaces of twisted quiver representations give different forms of the moduli space of quiver representations.