Lie Derivations of Dual Extension Algebras

TL;DR

This paper proves that all Lie derivations of the dual extension of a path algebra over a finite acyclic quiver are of standard form, clarifying the structure of derivations in this algebra class.

Contribution

It establishes that every Lie derivation of the dual extension algebra of a finite acyclic quiver's path algebra is of the standard form, a new structural result.

Findings

All Lie derivations are of standard form.

Clarifies the structure of derivations in dual extension algebras.

Provides a foundation for further algebraic investigations.

Abstract

Let $K$ be a field and $\Gamma$ a finite quiver without oriented cycles. Let $\Lambda$ be the path algebra $K(\Gamma, \rho)$ and let $\mathscr{D}(\Lambda)$ be the dual extension of $\Lambda$. In this paper, we prove that each Lie derivation of $\mathscr{D}(\Lambda)$ is of the standard form.