Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra
Rafael Reno S. Cantuba, Mark Anthony C. Merciales
TL;DR
The paper addresses Lie-polynomial characterization in $q$-deformed universal enveloping algebras of the 2D nonabelian Lie algebra, focusing on algebras $\mathcal{U}_q(r,0)$ and $\mathcal{U}_q(0,s)$. It develops a Diamond-Lemma-based framework to obtain explicit bases, derives crucial commutation identities, and establishes a concrete basis for $\mathcal{U}_q(r,s)$ in terms of words $B^lC^m$ and $C^mA^t$, with $C=[A,B]$. For the special cases $\mathcal{U}_q(r,0)$ and $\mathcal{U}_q(0,s)$, the authors characterize the Lie subalgebras $\mathcal{L}_{(r,0)}$ and $\mathcal{L}_{(0,s)}$, providing explicit bases and showing a Lie-algebra isomorphism $\mathcal{L}_{(s,0)} \cong \mathcal{L}_{(0,s)}$ via a sign-flipping map that interchanges $A$ and $B$. A key finding is that no nontrivial $q$-deformed associative homomorphisms exist between the corresponding algebras for $q\neq 1$, while the Lie-algebra structures can be isomorphic, highlighting a fundamental distinction between associative and Lie structures under $q$-deformation.
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.