The infinite dimensional Unital 3-Lie Poisson algebra

Abstract

From a commutative associative algebra A, the infinite dimensional unital 3-Lie Poisson algebra L is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of L is discussed. It is proved that: (1) there is a minimal set of generators S consisting of six vectors; (2) the quotient algebra L/FL 0 0,0 is a simple 3-Lie Poisson algebra;(3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra W 3 , A δ ω , A ω and the 3-W ∞ algebra can be embedded in L.