Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra

TL;DR

This paper investigates Lie polynomial characterizations within a $q$-deformed universal enveloping algebra of a two-dimensional nonabelian Lie algebra, providing solutions to generalized relations.

Contribution

It introduces solutions to Lie polynomial characterization problems in $q$-deformed universal enveloping algebras of a low-dimensional Lie algebra.

Findings

Solutions to generalized relations $AB-qBA=rA$ and $AB-qBA=sB$ are provided.

Characterization problems are solved for specific $q,r,s$ parameters.

The work extends understanding of Lie polynomials in deformed algebraic structures.

Abstract

The nonabelian two-dimensional Lie algebra over a field $\mathbb{F}$ has a presentation by generators $A$, $B$ and relation $\left[ A,B\right]=A$, with the universal enveloping algebra having a presentation by generators $A$, $B$ and relation $AB-BA=A$. A well-known fact is that the said Lie algebra is isomorphic to that which has a universal enveloping algebra that has a presentation by generators $A$, $B$ and relation $AB-BA=B$. Given $q,r,s\in\mathbb{F}$, solutions to the Lie polynomial characterization problems in the corresponding $q$-deformed universal enveloping algebras, with generalized relations $AB-qBA=rA$ and $AB-qBA=sB$, respectively, are presented.