Quantum generalized Heisenberg algebras and their representations

Lopes, Samuel A.; Razavinia, Farrokh

doi:10.1080/00927872.2021.1959602

Cited by 5 publications

(5 citation statements)

References 29 publications

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“…In [98] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras (qGHA for short), as they can be seen simultaneously as deformations and as generalizations of the generalized Heisenberg algebras appearing in [44] and profusely studied thenceforth in the Physics literature (see e.g. [45], [21], [9] and the references therein).…”

Section: Quantum Generalized Heisenberg Algebrasmentioning

confidence: 99%

“…In [98] we completely classify the finite-dimensional simple H q (f, g)-modules for all polynomials f, g ∈ F[h], assuming only that q ̸ = 0 and F is algebraically closed. In particular, this study generalizes and unifies the classification of finite-dimensional simple modules over down-up algebras, generalized down-up algebras and generalized Heisenberg algebras, which has been carried out over the series of references [19], [32], [33], [102], [126], [92] and [117], to name a few.…”

Section: 52mentioning

confidence: 99%

“…The main result in [98] is the following theorem. For the sake of brevity, we omit some of the definitions appearing below and refer the reader to [98,Sec. 3] for complete details.…”

Section: 52mentioning

confidence: 99%

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Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Lopes

2023

Communications in Mathematics

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Section: Quantum Generalized Heisenberg Algebrasmentioning

confidence: 99%

Section: 52mentioning

confidence: 99%

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Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Lopes

2023

Communications in Mathematics

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“…All these, and most probably many more, refer to H(q) = F ⟨A, B⟩ /(AB − qBA − 1) as the q-deformed Heisenberg algebra. No confusion should arise with similar symbol and terminology from recent studies like [20,21,22,23,24], which are not the subject of this paper.…”

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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“…The main example of these algebras is the one-dimensional central extension of a 2n-dimensional abelian Lie algebra, also named as the (2n+1)-dimensional Heisenberg algebra. Many generalizations of Heisenberg Lie algebras are under a certain consideration [24,27]. Let us call a 2-step nilpotent algebra with one-dimensional square a noncommutative Heisenberg algebra.…”

Section: Introductionmentioning

confidence: 99%