Local and 2-local Lie n-derivations of triangular algebras

Abstract

Let $\mathcal{R}$ be a commutative ring with identity, and $\mathcal{A},\mathcal{B}$ be unital algebras over $\mathcal{R}$. Let $\mathcal{M}$ be a unital $(\mathcal{A},\mathcal{B})$-bimodule, which is faithful as a left $\mathcal{A}$-module and also as a right $\mathcal{B}$-module. Let $\mathcal{T}=\mathit{\text{Tri}}(\mathcal{A},\mathcal{M},\mathcal{B})$ be an $(n-1)$-torsion free triangular algebra. In this note, we investigate the structure of local Lie n-derivations under certain mild conditions and 2-local Lie n-derivations under weaker conditions than that of local Lie n-derivations with $n\ge 3$ on $\mathcal{T}$. Under their corresponding conditions, we conclude that they are both standard on $\mathcal{T}$.

KEYWORDS:

• Derivation
• Lie n-derivation
• local and 2-local Lie n-derivations
• triangular algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 47L35

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions which helped in improving the quality of this article.

Additional information

Funding

The article is supported by Fundamental Research Program of Shanxi Province (No. 202103021224104) and Fundamental Research Program of Shanxi Province (No. 202103021223110) and Fundamental Research Program of Shanxi Province (No. 202103021223038) and the Scientific and Technological Innovation Programs of Higher Education Programs in Shanxi (No. 2021L015). The article is supported by Fundamental Research Program of Shanxi Province (No. 202103021224104) and Fundamental Research Program of Shanxi Province (No. 202103021223038) and the Scientific and Technological Innovation Programs of Higher Education Programs in Shanxi (No. 2021L015).