A Lie algebra related to the universal Askey-Wilson algebra

Abstract

Let $\mathbb{F}$ denote an algebraically closed field. Denote the three-element set by $\mathcal{X}=\{A,B,C\}$, and let $\mathbb{F}\left<\mathcal{X}\right>$ denote the free unital associative $\mathbb{F}$-algebra on $\mathcal{X}$. Fix a nonzero $q\in\mathbb{F}$ such that $q^4\neq 1$. The universal Askey-Wilson algebra $\Delta$ is the quotient space $\mathbb{F}\left<\mathcal{X}\right>/\mathbb{I}$, where $\mathbb{I}$ is the two-sided ideal of $\mathbb{F}\left<\mathcal{X}\right>$ generated by the nine elements $UV-VU$, where $U$ is one of $A,B,C$, and $V$ is one of \begin{equation} (q+q^{-1}) A+\frac{qBC-q^{-1}CB}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) B+\frac{qCA-q^{-1}AC}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) C+\frac{qAB-q^{-1}BA}{q-q^{-1}}.\nonumber \end{equation} Turn $\mathbb{F}\left<\mathcal{X}\right>$ into a Lie algebra with Lie bracket $\left[ X,Y\right] = XY-YX$ for all $X,Y\in\mathbb{F}\left<\mathcal{X}\right>$. Let $\mathcal{L}$ denote the Lie subalgebra of $\mathbb{F}\left<\mathcal{X}\right>$ generated by $\mathcal{X}$, which is also the free Lie algebra on $\mathcal{X}$. Let $L$ denote the Lie subalgebra of $\Delta$ generated by $A,B,C$. Since the given set of defining relations of $\Delta$ are not in $\mathcal{L}$, it is natural to conjecture that $L$ is freely generated by $A,B,C$. We give an answer in the negative by showing that the kernel of the canonical map $\mathbb{F}\left<\mathcal{X}\right>\rightarrow\Delta$ has a nonzero intersection with $\mathcal{L}$. Denote the span of all Hall basis elements of $\mathcal{L}$ of length $n$ by $\mathcal{L}_n$, and denote the image of $\sum_{i=1}^n\mathcal{L}_i$ under the canonical map $\mathcal{L}\rightarrow L$ by $L_n$. We study some properties of $L_4$ and $L_5$.