Classification of Transposed Poisson 3-Lie algebras of dimension 3

TL;DR

This paper classifies 3-dimensional transposed Poisson 3-Lie algebras by explicitly determining derivations and automorphisms of a unique algebra, providing a comprehensive understanding of their structure.

Contribution

It explicitly determines all derivations and automorphisms of a key 3-Lie algebra and classifies all 3-dimensional transposed Poisson 3-Lie algebras under specific conditions.

Findings

Explicit derivations and automorphisms of the 3-dimensional algebra.

Complete classification of transposed Poisson 3-Lie algebras of dimension 3.

Identification of isomorphism classes under certain triviality conditions.

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 $\mathbb{C}$.