Transposed Poisson Structures on Two Nambu 3-Lie Algebras

Abstract

We describe the $\frac{1}{3}$-derivations and transposed Poisson structures of the Nambu 3-Lie algebras $A_{\ddot{I}}{‰}^{\hat{I}}´$ and $A_{f,k}$. Specifically, we first present that $A_{\ddot{I}}{‰}^{\hat{I}}´$ is finitely generated and graded. Then we find that $A_{\ddot{I}}{‰}^{\hat{I}}´$ 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.