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arXiv · 2502.16074

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

Rafael Reno S. Cantuba·Mark Anthony C. Merciales
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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.

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Rafael Reno S. Cantuba, Mark Anthony C. Merciales. 2025-02-22. Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra. https://doi.org/10.24330/ieja.1823898

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