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

Abstract

We introduce a dual notion of the Poisson algebra, called the transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. The transposed Poisson algebra shares common properties of the Poisson algebra and arises naturally 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 algebra with two commuting derivations similarly applies to a transposed Poisson algebra. More broadly, the transposed Poisson algebra 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 that produce 3-Lie algebras from Poisson algebras with a strongness condition.