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

TL;DR

This paper constructs a basis for free GDN-Poisson algebras, introduces special admissible algebras, and proves embedding and isomorphism results, advancing the understanding of their algebraic structure and relations to differential algebras.

Contribution

It provides a basis construction for free GDN-Poisson algebras and establishes their embeddings into special admissible algebras, linking them to differential algebra structures.

Findings

Constructed a linear basis for free GDN-Poisson algebras.

Proved that any GDN-Poisson algebra embeds into its universal enveloping algebra.

Showed that certain GDN-Poisson algebras are isomorphic to commutative differential algebras.

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.