On free Gelfand--Dorfman--Novikov superalgebras and a PBW type theorem

TL;DR

This paper constructs a basis for free GDN superalgebras over fields with characteristic not equal to 2, proves a PBW theorem for embedding these superalgebras into their universal enveloping algebras, and discusses an Engel theorem under certain conditions.

Contribution

It introduces a basis for free GDN superalgebras and establishes a PBW theorem, extending classical results to the superalgebra context.

Findings

Constructed a linear basis for free GDN superalgebras.

Proved a PBW theorem for GDN superalgebras.

Provided an Engel theorem under specific assumptions.

Abstract

We construct a linear basis of a free GDN superalgebra over a field of characteristic $\neq 2$. As applications, we prove a PBW theorem, that is, any GDN superalgebra can be embedded into its universal enveloping commutative associative differential superalgebra. An Engel theorem under some assumptions is given.