Simple weight modules over weak generalized Weyl algebras

Lü, Rencai; Mazorchuk, Volodymyr; Zhao, Kaiming

doi:10.1016/j.jpaa.2014.12.003

Cited by 9 publications

(5 citation statements)

References 19 publications

(24 reference statements)

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“…[45], [21], [9] and the references therein). In the Mathematics literature, generalized Heisenberg algebras were studied mainly in [102], [101] and [96]. For an overview of their relevance in mathematical Physics see the introductory section in [102].…”

Section: Quantum Generalized Heisenberg Algebrasmentioning

confidence: 99%

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Lopes

2023

Communications in Mathematics

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Section: Quantum Generalized Heisenberg Algebrasmentioning

confidence: 99%

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Lopes

2023

Communications in Mathematics

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“…Not included in this diagram is the notion of a (rank one) weak generalized Weyl algebra [77]. In this setting, the map σ is only required to be endomorphism.…”

Section: Future Work and Research Directionsmentioning

confidence: 99%

Fixed rings of generalized Weyl algebras

Gaddis

Won

2019

Journal of Algebra

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Generalized Weyl Algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings, including global dimension, rigidity, and simplicity. IntroductionThroughout, k is a field and all algebras are associative k-algebras.The Shephard-Todd-Chevalley (STC) Theorem [6,24] gives conditions for the fixed ring of a polynomial ring by a finite group of linear automorphisms to again be a polynomial ring. More recently, there has been significant interest in studying STC-like theorems in noncommutative algebra, in particular whether the fixed ring of an (N-graded) Artin-Schelter regular algebra again has this property [15].To consider this problem outside of the N-graded setting, one could ask whether the fixed ring of a (twisted) Calabi-Yau algebra is again (twisted) Calabi-Yau. Algebras satisfying this property have attracted much interest of late [9,18,20,21]. Since polynomial rings are (trivially) Calabi-Yau, this is indeed a reasonable generalization.An important family of Z-graded (twisted) Calabi-Yau algebras are the generalized Weyl algebras (GWAs) and so they serve as a good test case of the STC question in this setting. Kirkman and Kuzmanovich [16] have proposed a version of the STC Theorem for GWAs, essentially asking when the fixed ring of a GWA again has GWA structure. We propose a strengthening of this: to determine when the fixed ring of a Calabi-Yau GWA is again a Calabi-Yau GWA.Generalized Weyl algebras were named by Bavula [2]. They have been studied extensively by many authors prior to and post Bavula's definition. Notably, the interested reader is directed to the work of Hodges [10, 11], Jordan [12], Joseph [14], Rosenberg [23], Smith [26], and Stafford [27]. There are severalimportant families of algebras that may be constructed as GWAs. In the classical case, this includes the Weyl algebras and primitive quotients of U (sl 2 ). We will primarily be concerned with a subclass known as quantum GWAs, which includes quantum planes, quantum Weyl algebras, and primitive quotients of U q (sl 2 ).The proposal of Kirkman and Kuzmanovich [16] mentioned above has its basis in the work of Jordan and Wells [13] and their study of fixed rings of GWAs by automorphisms that fix the base ring. Won and the first-named author showed that it is possible to diagonalize any filtered automorphism of a classical GWA 1 with quadratic defining polynomial so that the Jordan and Wells result may be applied [8]. The methods and results in this paper vary wildly from these aforementioned works.Results. First, in Section 2, we study automorphisms of quantum GWAs. We use the classification of their automorphism groups by Suárez-Alvarez and Vivas [28] in order to classify their finite subgroups (Proposition 2.6). In Section 3 w...

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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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Simple weight modules over weak generalized Weyl algebras

Cited by 9 publications

References 19 publications

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Fixed rings of generalized Weyl algebras

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

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