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 $\mathbb{F}⟨\mathcal{X}⟩$ denote the free unital associative $\mathbb{F}$-algebra on $\mathcal{X}$. Fix a nonzero $q\in \mathbb{F}$ such that $q^4\ne 1$. The universal Askey-Wilson algebra $\Delta$ is the quotient space $\mathbb{F}⟨\mathcal{X}⟩/\mathbb{I}$, where $\mathbb{I}$ is the two-sided ideal of $\mathbb{F}⟨\mathcal{X}⟩$ 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 $\mathbb{F}⟨\mathcal{X}⟩$ into a Lie algebra with Lie bracket $\left[X,Y\right]=XY-YX$ for all $X,Y\in \mathbb{F}⟨\mathcal{X}⟩$. Let $\mathcal{L}$ denote the Lie subalgebra of $\mathbb{F}⟨\mathcal{X}⟩$ 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 $\mathbb{F}⟨\mathcal{X}⟩\to \Delta$ 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 $\mathcal{L}\to L$ by $L_n$. We study some properties of $L_4$ and $L_5$.

References in corpus (1)

• The Universal Askey-Wilson Algebra

Cited by in corpus (4)

• Lie polynomials in $q$-deformed Heisenberg algebras
• Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials
• Lie polynomials in an algebra defined by a linearly twisted commutation relation
• Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra