The Coupled q-oscillator Algebra and a Classification of Its Simple Weight Modules

Abstract

The coupled q-oscillator algebra ${\cal A}$ is a smash product of the polynomial algebra in one variable with the Hopf algebra $U_{q}({\mathfrak s}{\mathfrak l}_{2})$. We show that the centre of the algebra ${\cal A}$ is trivial. We find a distinguished normal element of ${\cal A}$ that plays a role in studying its structures and representations. We give explicit descriptions of the prime, completely prime, primitive and maximal ideals of the algebra ${\cal A}$. As a result, we show that ${\cal A}$ cannot have a Hopf algebra structure, and ${\cal A}$ has no finite-dimensional representations. The group of automorphisms of ${\cal A}$ is explicitly described. A classification of all simple weight modules over the algebra ${\cal A}$ is obtained. Using the classification of primitive ideals of ${\cal A}$, we determine the annihilator of every simple weight module.

Keywords

Quantum algebra / prime ideal / automorphism group / weight module / 16T20 / 16S40 / 17B37 / 16D60

References

[1] BavulaVV. Generalized Weyl algebras and their representations. St. Petersburg Math. J., 1993, 4(1): 71-92

[2] BavulaVV. Filter dimension of algebras and modules, a simplicity criterion of generalized Weyl algebras. Comm. Algebra, 1996, 24(6): 1971-1992.

[3] BavulaVV. Classification of the simple modules of the quantum Weyl algebra and the quantum plane. Quantum Groups and Quantum Spaces (Warsaw, 1995), 1997, Warsaw. Polish Acad. Sci. Inst. Math.. 19320140

[4] BavulaVV, LuT. The prime spectrum and simple modules over the quantum spatial ageing algebra. Algebr. Represent. Theory, 2016, 19(5): 1109-1133.

[5] BavulaVV, LuT. The prime spectrum of the algebra ${\mathbb K}_{q}[X,Y] \rtimes U_{q}({\mathfrak s}{\mathfrak l}_{2})$ and a classification of simple weight modules. J. Noncommut. Geom., 2018, 12(3): 889-946.

[6] BavulaVV, LuT. Classification of simple weight modules over the Schrödinger algebra. Canad. Math. Bull., 2018, 61(1): 16-39.

[7] BergenJ. Taft algebras acting on associative algebras. J. Algebra, 2015, 423: 422-440.

[8] BonatsosD, DaskaloyannisC. Quantum groups and their applications in nuclear physics. Progress in Particle and Nuclear Physics, 1999, 43: 537-618.

[9] BrownKA, GoodearlKRLectures on Algebraic Quantum Groups, 2002, Basel. Birkhäuser. .

[10] BužekV. The Jaynes–Cummings model with a q analogue of a coherent state. J. Modern Opt., 1992, 39(5): 949-959.

[11] ChaichianM, EllinasD, KulishP. Quantum algebra as the dynamical symmetry of the deformed Jaynes-Cummings model. Phys. Rev. Lett., 1990, 65(8): 980-983.

[12] ChaichianM, Gonzales FelipeR, MontonenC. Statistics of q-oscillators, quons and relations to fractional statistics. J. Phys. A, 1993, 26(16): 4017-4034.

[13] DingF, TsymbaliukA. Representations of infinitesimal Cherednik algebras. Represent. Theory, 2013, 17: 557-583.

[14] DuplijS, SineľshchikovSS. Classification of $U_{q}({\mathfrak s}{\mathfrak l}_{2})$-module algebra structures on the quantum plane. J. Math. Phys. Anal. Geom., 2010, 6(4): 406-430436, 439

[15] EtingofP, GanW, GinzburgV. Continuous Hecke algebras. Tansform. Groups, 2005, 10(3–4): 423-447.

[16] EtingofP, GinzburgV. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math., 2002, 147(2): 243-348.

[17] FarinatiM. Hochschild duality, localization, and smash products. J. Algebra, 2005, 284(1): 415-434.

[18] GanWL, KhareA. Quantized symplectic oscillator algebra of rank one. J. Algebra, 2007, 310(2): 671-707.

[19] GoodearlKR. Prime ideals in skew polynomial rings and quantized Weyl algebras. J. Algebra, 1992, 150(2): 324-377.

[20] JordanDA. Normal elements of degree one in Ore extensions. Comm. Algebra, 2002, 30(2): 803-807.

[21] KinserR, WaltonC. Actions of some pointed Hopf algebras on path algebras of quivers. Algebra Number Theory, 2016, 10(1): 117-154.

[22] KlimykA, SchmüdgenKQuantum Groups and Their Representations, 1997, Berlin. Springer-Verlag. .

[23] Le MeurP. Smash products of Calabi-Yau algebras by Hopf algebras. J. Noncommut. Geom., 2019, 13(3): 887-961.

[24] LüR, MazorchukV, ZhaoK. Classification of simple weight modules over the 1-spatial ageing algebra. Algebr. Represent. Theory, 2015, 18(2): 381-395.

[25] MajidS. Braided momentum in the q-Poincaré group. J. Math. Phys., 1993, 34(5): 2045-2058.

[26] MathieuO. Classification of irreducible weight modules. Ann. Inst. Fourier (Grenoble), 2000, 50(2): 537-592.

[27] McConnellJC, RobsonJCNoncommutative Noetherian Rings, 2001, Providence, RI. American Mathematical Society. 30

[28] MontgomerySHopf Algebras and Their Actions on Rings, 1993, Providence, RI. American Mathematical Society. .

[29] MontgomeryS, SmithSP. Skew derivations and Uq(sl(2)). Israel J. Math., 1990, 72(1–2): 158-166.

[30] NeškovićPV, UroševićBV. Quantum oscillators: applications in statistical mechanics. Internat. J. Modern Phys. A, 1992, 7(14): 3379-3388.

[31] TaoW. The Heisenberg double of the quantum Euclidean group and its representations. Sci. China Math., 2023, 66(8): 1713-1736.

RIGHTS & PERMISSIONS

Peking University