Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator

Cantuba, Rafael Reno S.

doi:10.1007/s11005-020-01304-x

Cited by 4 publications

(8 citation statements)

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“…Similar studies were done for q-deformed Heisenberg algebras [3,4]. In all such studies [2,3,4] the focus was on the consequences of the non-Lie polynomial, deformed commutation relations on the Lie polynomials in the same algebra. The studies [2,3,4] have motivated further progress as reported in [5,6], in which central extensions and torsion-type deformation parameters were considered, respectively.…”

Section: Introductionmentioning

confidence: 93%

“…the related family of Askey-Wilson algebras are not Lie polynomials in the generators [2], and so the usual algebraic machinery for finitely generated and finitely presented Lie algebras is not directly applicable, but it was further shown that the Lie subalgebra generated by the generators of the algebra is not free [2]. Similar studies were done for q-deformed Heisenberg algebras [3,4]. In all such studies [2,3,4] the focus was on the consequences of the non-Lie polynomial, deformed commutation relations on the Lie polynomials in the same algebra.…”

Section: Introductionmentioning

confidence: 99%

“…In all such studies [2,3,4] the focus was on the consequences of the non-Lie polynomial, deformed commutation relations on the Lie polynomials in the same algebra. The studies [2,3,4] have motivated further progress as reported in [5,6], in which central extensions and torsion-type deformation parameters were considered, respectively. A survey of the particular type of Lie structure, in associative algebras being described here, can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

“…The approach in determining a basis for L q done here is something that the author realized because of the works [3,4] that he did on q-deformed Heisenberg algebras.…”

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confidence: 99%

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A Casimir Element Inexpressible as a Lie Polynomial

Cantuba

2021

International Electronic Journal of Algebra

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Let q be a scalar that is not a root of unity. We show that any nonzero polynomial in the Casimir element of the Fairlie-Odesskii algebra U q (so 3 ) cannot be expressed in terms of only Lie algebra operations performed on the generators I 1 , I 2 , I 3 in the usual presentation of U q (so 3 ). Hence, the vector space sum of the center of U q (so 3 ) and the Lie subalgebra of U q (so 3 ) generated by I 1 , I 2 , I 3 is direct.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The approach in determining a basis for L q done here is something that the author realized because of the works [3,4] that he did on q-deformed Heisenberg algebras.…”

mentioning

confidence: 99%

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A Casimir Element Inexpressible as a Lie Polynomial

Cantuba

2021

International Electronic Journal of Algebra

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

Cantuba

2021

J. Algebra Appl.

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We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

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Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator

Cited by 4 publications

References 22 publications

A Casimir Element Inexpressible as a Lie Polynomial

A Casimir Element Inexpressible as a Lie Polynomial

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

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

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