Classification of Transposed Poisson 3-Lie algebras of dimension 3

TL;DR

This work addresses the classification problem for transposed Poisson $3$-Lie algebras in dimension three, focusing on the unique nontrivial complex $3$-Lie algebra $(A_3,[\cdot,\cdot,\cdot])$ with $[e_1,e_2,e_3]=e_1$ and the case where the left multiplication $L_{e_1}$ is trivial. The authors compute all $\frac{1}{3}$-derivations and automorphisms of $(A_3,[\cdot,\cdot,\cdot])$ and derive the corresponding transposed Poisson $3$-Lie algebra structures, revealing $26$ candidate algebras $T_i'$. They then classify these up to automorphisms, showing that only $10$ isomorphism classes, $T_1,\dots,T_{10}$, remain for $L_{e_1}=0$ and provide explicit commutative multiplications for these classes. The proofs are organized through four case analyses that identify, relate, and distinguish the resulting algebras, yielding a complete dimension-three classification under the stated constraint and contributing to the broader understanding of transposed Poisson $3$-Lie structures.

Abstract

Transposed Poisson $3$-Lie algebra is a dual notion of Nambu-Poisson algebra of order 3. In this paper, we explicitly determine all $\frac{1}{3}$-derivations and automorphisms of the unique nontrivial $3$-dimensional complex $3$-Lie algebra $(A_3,[\cdot,\cdot,\cdot])$. Based on the one-one correspondence between $\frac{1}{3}$-derivations and transposed Poisson 3-Lie algebras, up to isomorphism, we classify transposed Poisson $3$-Lie algebras of dimension $3$ under the case that $L_{e_1}$ is trivial over the complex field $\mathbb{C}$.

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 3.1
• proof
• Proposition 3.2
• proof