The infinite dimensional unital Nambu–Poisson algebra of order 3

Abstract

From a commutative associative algebra with a basis 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 consisting of six vectors; (2) the quotient 3-Lie algebra is simple; (3) four infinite dimensional 3-Lie algebras: the 3-Virasoro–Witt algebra (), and the 3-algebra can be embedded in the unital Nambu–Poisson algebra of order 3.