Lie polynomials in a $q$-deformed universal enveloping algebra of the two-dimensional non-abelian Lie algebra

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 solution to the Lie polynomial characterization problem in the corresponding class of $q$-deformed universal enveloping algebras, specifically of the algebra with relation $AB-qBA=A$ is presented.

Keywords

• Lie polynomial
• $q$-analog
• universal enveloping algebra
• Lie algebra

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