Transposed Poisson Structures on Two Nambu 3-Lie Algebras

TL;DR

This paper investigates the algebraic structures called transposed Poisson structures on specific Nambu 3-Lie algebras, revealing their derivations, gradings, and the existence of non-trivial structures.

Contribution

It characterizes the transposed Poisson structures and derivations of two particular Nambu 3-Lie algebras, providing new insights into their algebraic properties.

Findings

$A_ ext{omega}^ ext{delta}$ has non-trivial 1/3-derivations but only trivial transposed Poisson structures.

$A_{f,k}$ admits non-trivial transposed Poisson structures.

$A_ ext{omega}^ ext{delta}$ is finitely generated and graded.

Abstract

We describe the $\frac{1}{3}$-derivations and transposed Poisson structures of the Nambu 3-Lie algebras $A_\omega^\delta $ and $ A_{f,k} $. Specifically, we first present that $A_\omega^\delta $ is finitely generated and graded. Then we find that $A_\omega^\delta $ has non-trivial $\frac{1}{3}$-derivations and admits only trivial transposed Poisson structures. The 3-Lie algebra $A_{f,k}$ admits non-trivial transposed Poisson structures.