Lie superderivations on unital algebras with idempotents

Abstract

Let $\mathcal{U}$ be an associative unital algebra containing a non-trivial idempotent e. We consider $\mathcal{U}$ as a superalgebra whose ${\mathbb{Z}}_2$-grading is induced by e. This paper aims to describe Lie superderivations of $\mathcal{U}$. In particular, we characterize the general form of Lie superderivations of $\mathcal{U}$ and apply it to present the necessary and sufficient conditions for a Lie superderivation on $\mathcal{U}$ to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their ${\mathbb{Z}}_2$-grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras.

KEYWORDS:

• Lie superderivation
• superalgebra
• superderivation
• triangular algebra
• unital algebra with non-trivial idempotents

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A70
• 17B60
• 16W25
• 15A78

Acknowledgments

The authors would like to express their sincere thanks to the referee(s) for careful reading the paper and useful suggestions.