Representations via differential algebras and equationally Noetherian algebras

TL;DR

This paper demonstrates that certain free algebras related to Witt, left-symmetric Witt, and symplectic Poisson algebras can be embedded into differential polynomial algebras, proving their equational Noetherian property.

Contribution

It introduces new representations of these algebras as subalgebras of differential polynomial algebras and establishes their equational Noetherianity.

Findings

Free algebras are subalgebras of differential polynomial algebras.

Witt, left-symmetric Witt, and symplectic Poisson algebras are equationally Noetherian.

The free algebras generated by these varieties are also equationally Noetherian.

Abstract

We show that free algebras of the variety of algebras generated by the Witt algebra $W_n$, the left-symmetric Witt algebra $L_n$, and the symplectic Poisson algebra $P_n$ can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that $W_n$, $L_n$, $P_n$, and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian.