Identities for a parametric Weyl algebra over a ring

TL;DR

This paper generalizes the construction of a parametric Weyl algebra over a ring and characterizes its polynomial identities under specific conditions, expanding understanding of algebraic identities in this context.

Contribution

It extends the algebraic framework of Weyl algebras to rings and describes their polynomial identities, a novel generalization of prior work.

Findings

Polynomial identities for the generalized algebra are characterized.

Conditions on the polynomial h influence the identities.

The work broadens the algebraic understanding of Weyl-type structures.

Abstract

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra $\A_h$ generated by elements $x,y,$ which satisfy the relation $yx-xy=h$ for some $0\neq h\in \FF[x]$. We generalize this construction to $\A_h(\B)$ by working over the fixed $\FF$-algebra $\B$ instead of $\FF$. We describe the polynomial identities for $\A_h(\B)$ over the infinite field $\FF$ in case $h\in\B[x]$ satisfies certain restrictions.