The infinite dimensional unital Nambu–Poisson algebra of order 3

Abstract

From a commutative associative algebra $(A,·)$ with a basis $\{L_{l,m}^r | r,l,m\in \frac{1}{2}\mathbb{Z}\},$ the infinite dimensional unital Nambu–Poisson algebra of order 3 is constructed, which is also a canonical Nambu–Poisson algebra, and its structures and derivations are discussed. It is proved that: (1) there is a minimal set of generators of $(A,·)$ consisting of six vectors; (2) the quotient 3-Lie algebra $\mathfrak{A}/\mathbb{F}{L}_{0,0}^0$ is simple; (3) four infinite dimensional 3-Lie algebras: the 3-Virasoro–Witt algebra ${\mathcal{W}}_3$ ($z=\pm 2\sqrt{-1}$), $A_{\omega }^{\delta },A_{\omega }$ and the $W_{\infty }$ 3-algebra can be embedded in the unital Nambu–Poisson algebra of order 3.

Keywords:

• 3-Lie algebra
• canonical Nambu–Poisson algebra
• Nambu–Poisson algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B05
• 17D99

Acknowledgments

We give our warmest thanks to the referees for very helpful suggestions that improve the paper.

Additional information

Funding

Ruipu Bai was supported by the Natural Science Foundation of Hebei Province, China (A2018201126).