Characterization of $n$-Lie Derivations on Generalized Matrix Algebras

TL;DR

The paper addresses the problem of describing $n$-Lie derivations on generalized matrix algebras for $n\ge 3$ by proving a structural decomposition. Using induction on $n$ with a base case for $n=3$, it shows that under mild center- and central-ideal conditions on the Morita-context algebra $\mathcal{G}= [AMNB]$, every $n$-Lie derivation $\varphi$ can be written as $\varphi=\kappa+\psi$, where $\kappa$ is an extremal $n$-derivation given by $\kappa(x_1,\dots,x_n)= [x_1,[x_2,[\cdots,[x_n, X_0]]\cdots]]$ with $X_0= e\varphi(e,\dots,e)f+(-1)^n f\varphi(e,\dots,e)e$, and $\psi$ is an $n$-linear centrally-valued mapping. This decomposition is then extended to general $n$ via induction and applied to key algebraic settings: full matrix algebras $\mathcal{M}_{r}(\mathcal{R})$ with $r>2$ and triangular algebras. The results unify the understanding of multilinear Lie-type derivations on generalized matrix algebras and provide explicit forms that facilitate further study of related maps such as biderivations and central mappings. Overall, the work advances multilinear derivation theory on Morita-context algebras and yields practical characterizations for canonical algebra families.

Abstract

The principal objective of this paper is to determine the structure of $n$-Lie derivations ($n\geq 3$) on generalized matrix algebras.It is shown that under certain mild assumptions, every $n$-Lie derivation can be decomposed into the sum of an extremal $n$-derivation and an $n$-linear centrally-valued mapping. As direct applications, we provide complete characterizations of $n$-Lie derivations on both full matrix algebras and triangular algebras.

Theorems & Definitions (39)

• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Proposition 3.5
• Proposition 3.6
• proof
• Lemma 3.7
• proof
• Remark 3.8