Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra
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2019
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| author | Cantuba, Rafael Reno S. |
| author_facet | Cantuba, Rafael Reno S. |
| contents | Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{ωN}$ for some nonzero real parameter $ω$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_03356 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra Cantuba, Rafael Reno S. Rings and Algebras Quantum Algebra 47L30, 17B60, 17B65, 16S15, 81R50 Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{ωN}$ for some nonzero real parameter $ω$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$. |
| title | Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra |
| topic | Rings and Algebras Quantum Algebra 47L30, 17B60, 17B65, 16S15, 81R50 |
| url | https://arxiv.org/abs/1910.03356 |