Equivalent definitions of the preprojective algebra

TL;DR

This paper demonstrates the equivalence of three different algebraic definitions of preprojective algebras for Dynkin quivers, using homological algebra techniques to unify the concepts.

Contribution

It introduces a new proof of the equivalence of definitions of preprojective algebras, extending Ringel's work with modern homological algebra methods.

Findings

Proves the equivalence of three definitions of preprojective algebras

Shows the standard and generalized commutator definitions are equivalent

Provides a new proof applying Happel's theorem and homological algebra techniques

Abstract

Following the article of C. M. Ringel we introduce preprojective algebras of a Dynkin quiver $Q$ starting from three definitions which, despite concerning completely different algebraic structures, turn out to be equivalent. Our main result is a new version of Ringel's proofs that applies a theorem by Happel and exploits the techniques of homological algebra. Moreover we show that the definition of the preprojective algebra given with the usual notion of commutator is equivalent to the definition with the "generalised" commutator.