# An extension of a $q$-deformed Heisenberg algebra and its Lie polynomials

## TL;DR

This paper extends the $q$-deformed Heisenberg algebra by adding a relation making the commutator central, and characterizes all Lie polynomials within this extended algebra.

## Contribution

It introduces a new algebra $
(q)$ with a central commutator and classifies all Lie polynomials in this algebra.

## Key findings

- Identified all Lie polynomials in the extended algebra.
- Extended the $q$-deformed Heisenberg algebra with a central commutator.
- Provided a complete description of the Lie polynomial structure.

## Abstract

Let $\mathbb{F}$ be a field, and fix a $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra over $\mathbb{F}$ with generators $A$, $B$ and a relation which asserts that $AB - qBA$ is the multiplicative identity in $\mathcal{H}(q)$. We extend $\mathcal{H}(q)$ into an algebra $\mathcal{R}(q)$ defined by generators $A$, $B$ and a relation which asserts that $AB-qBA$ is central in $\mathcal{R}(q)$. We identify all elements of $\mathcal{R}(q)$ that are Lie polynomials in $A$, $B$.