# Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials

## TL;DR

This paper characterizes Lie polynomials in the $q$-deformed Heisenberg algebra when $q$ is a root of unity, exploring its algebraic structure and related Lie algebras.

## Contribution

It provides a complete characterization of Lie polynomials in torsion-type $q$-deformed Heisenberg algebras and analyzes their basis, grading, and commutation relations.

## Key findings

- Characterization of all Lie polynomials in torsion-type $ ext{Heisenberg algebra}$
- Determination of basis and graded structure of associated Lie algebras
- Analysis of commutation relations within these Lie algebras

## Abstract

Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative identity. For $\mathcal{H}(q)$ of torsion-type, that is if $q$ is a root of unity, characterization is obtained for all the Lie polynomials in $A,B$ and basis and graded structure and commutation relations for associated Lie algebras are studied.