Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras

Abstract

We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common properties of the Poisson algebra, including the closure under taking tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic construction of a Poisson algebra from a commutative associative algebra with a pair of commuting derivations has a similar construction of a transposed Poisson algebra when there is one derivation. More broadly, the transposed Poisson algebra also captures the algebraic structures when the commutator is taken in pre-Lie Poisson algebras and two other Poisson type algebras. Furthermore, the transposed Poisson algebra improves two processes in~[17] that produce 3-Lie algebras from Poisson algebras with a strongness condition. When transposed Poisson algebras are used in one process, the strongness condition is no longer needed and the resulting 3-Lie algebra gives a transposed Poisson 3-Lie algebra. In the other process, the resulting 3-Lie algebra is shown to again give a transposed Poisson 3-Lie algebra.