On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem

Abstract

In 1997, X. Xu \cite{Xiaoping Xu Poisson} invented a concept of Novikov-Poisson algebras (we call them Gelfand-Dorfman-Novikov-Poisson (GDN-Poisson) algebras). We construct a linear basis of a free GDN-Poisson algebra. We define a notion of a special GDN-Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see \cite{Gelfand}). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN-Poisson algebra is embeddable into its universal enveloping special GDN-Poisson admissible algebra. Also we prove that any GDN-Poisson algebra with the identity $x\circ (y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y$ is isomorphic to a commutative associative differential algebra.

23 pages

Cited by in corpus (11)

• Gelfand--Dorfman algebras, derived identities, and the Manin product of operads
• On the special identities of Gelfand--Dorfman algebras
• Polynomial identities in Novikov algebras
• Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras
• Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras
• One-generated nilpotent Novikov algebras
• The algebraic classification of nilpotent Novikov algebras
• On radicals of Novikov algebras
• On free Gelfand--Dorfman--Novikov superalgebras and a PBW type theorem
• Distributive lattices of varieties of Novikov algebras
• Invariant theory of relatively free right-symmetric and Novikov algebras