Transposed Poisson Structures on Two Nambu 3-Lie Algebras
Jingjing Jiang, Chunyi Li, Jie Lin
TL;DR
The paper addresses transposed Poisson structures on two canonical Nambu 3-Lie algebras, $A_{ ame{ extomega}}^{ ame{ extdelta}}$ and $A_{f,k}$, linking them to $ rac{1}{3}$-derivations. It shows that $A_{ ame{ extomega}}^{ ame{ extdelta}}$ is finitely generated and $oxed{ ext{Z}}$-graded with homogeneous $ rac{1}{3}$-derivations $D_k$, and that all transposed Poisson structures on this algebra are trivial, while $A_{f,k}$ admits nontrivial transposed Poisson structures characterized by parameters $(oldsymbol{ extalpha}, f, c_p, d_{i,j,q})$ subject to explicit relations. A concrete construction is provided for $A_{f,k}$, illustrating how the grading and derivation structure constrain possible multiplications. These results deepen the understanding of transposed Poisson structures in canonical Nambu 3-Lie algebras and highlight how graded derivations govern feasible algebraic compatibilities.
Abstract
We describe the $\frac{1}{3}$-derivations and transposed Poisson structures of the Nambu 3-Lie algebras $A_ω^δ$ and $ A_{f,k} $. Specifically, we first present that $A_ω^δ$ is finitely generated and graded. Then we find that $A_ω^δ$ 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.