On Trivial Extensions and Higher Preprojective Algebras

TL;DR

This paper establishes a deep connection between twisted trivial extensions and higher preprojective algebras for Koszul n-homogeneous algebras, extending classical correspondences in noncommutative algebra.

Contribution

It demonstrates that the quadratic dual of certain twisted trivial extensions is isomorphic to higher preprojective algebras, providing a new perspective on algebraic dualities.

Findings

Quadratic dual of twisted trivial extension equals higher preprojective algebra

Application to τ-slice algebras of stable n-translation algebras

Noncommutative Bernstein-Gelfand-Gelfand correspondence

Abstract

In this paper, we show that for a Koszul $n$-homogeneous algebra $\Lambda$, the quadratic dual of certain twisted trivial extension is the $(n+1)$-preprojective algebra of its quadratic dual, that is, $ (\Delta_{\nu}\Lambda)^{!,op} \simeq\Pi( \Lambda^{ !, op })$. This is applied to the $\tau$-slice algebras of stable $n$-translation algebras and gives a noncommutative version of Bernstein-Gelfand-Gelfand correspondence for such algebras.