Tilting theoretical approach to moduli spaces over preprojective algebras

TL;DR

This paper explores the use of tilting theory over preprojective algebras to establish equivalences between categories of semistable modules and demonstrate isomorphisms between their moduli spaces, including cases related to Kleinian singularities.

Contribution

It introduces reflection functors via tilting modules that induce isomorphisms between moduli spaces, generalizing McKay correspondence results.

Findings

Established equivalences between categories of semistable modules

Proved isomorphisms between moduli spaces induced by tilting theory

Extended results to moduli spaces related to Kleinian singularities

Abstract

We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using tilting modules over $Lambda$. Moreover we prove that the equivalence induces an isomorphism of algebraic varieties between moduli spaces. In particular, we study in the case when the moduli spaces related to the Kleinian singularity. We generalize a result of Crawley-Boevey which is known another proof of the McKay correspondence of Ito-Nakamura type.