Lie polynomials in an algebra defined by a linearly twisted commutation relation

Cantuba, Rafael Reno S.

doi:10.1142/s0219498822501754

Cited by 1 publication

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“…Some works published after [16] continue to refer to F ⟨A, B⟩ /(AB − qBA − 1) as H(q) or as the q-deformed Heisenberg algebra. Said works come from varied fields of mathematics, such as Ring Theory [15,17], Lie algebras [6,8,9,10,11], Mathematical Physics [7,19], and algebraic curves [13]. All these, and most probably many more, refer to H(q) = F ⟨A, B⟩ /(AB − qBA − 1) as the q-deformed Heisenberg algebra.…”

Section: Introductionmentioning

confidence: 99%

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Cantuba¹

2023

Communications in Mathematics

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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 $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

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