On deformed preprojective algebras

TL;DR

This paper explores deformed preprojective algebras, establishing their 2-Calabi-Yau property, unifying reflection functors, and classifying tilting ideals, thereby advancing understanding in noncommutative geometry and related fields.

Contribution

It introduces the 2-Calabi-Yau property for deformed preprojective algebras, unifies reflection functors, and classifies tilting ideals in these algebras.

Findings

Proved deformed preprojective algebras are 2-Calabi-Yau.

Unified reflection functors for different algebra types.

Classified tilting ideals in 2-Calabi-Yau algebras.

Abstract

Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed preprojective algebras for extended Dynkin quivers.