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Whittaker Modules Over Some Generalized Weyl Algebras

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Abstract

In Benkart and Ondrus (Represent. Theory 13, 141–164 2009), Benkart and Ondrus investigated Whittaker modules for generalized Weyl algebras. Following the results of Benkart and Ondrus, we study Whittaker modules for three special kinds of generalized Weyl algebras in this note, including Rueda’s algebras, the algebras Uq(f(K)) and Uq(f(K,H)). In particular, we acquire the centers of the last two classes of algebras before giving an explicit description of their simple Whittaker modules.

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References

  1. Adamović, D., Lü, R., Zhao, K.: Whittaker modules for the affine Lie algebra \(\mathit {A}^{(1)}_{1}\). Adv. Math. 289, 438–479 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bavula, V.: Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules. In: Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), 23 pp. Carleton-Ottawa Math. Lecture Note Ser. 14, Carleton Univ., Ottawa, ON (1992)

  3. Bavula, V.: Generalized Weyl algebras and their representations. (Russian) Algebra i Analiz. 4(1), 75–95 (1992); translation in St. Petersburg Math. J. 4(1), 71–92. (1993)

  4. Benkart, G.: Multiparameter Weyl algebras. arXiv:1306.0485(2013)

  5. Benkart, G., Ondrus, M.: Whittaker modules for generalized Weyl algebras. Represent. Theory 13, 141–164 (2009)

    Article  MathSciNet  Google Scholar 

  6. Cai, Y., Cheng, Y., Shen, R.: Quasi-Whittaker modules for the Schrödinger algebra. Linear Algebra Appl. 463, 16–32 (2014)

    Article  MathSciNet  Google Scholar 

  7. Cai, Y., Shen, R., Zhang, J.: Whittaker modules and quasi-Whittaker modules for the Euclidean Lie algebra \(\mathfrak {e}~(3)\). J. Pure Appl. Algebra 220(4), 1419–1433 (2016)

    Article  MathSciNet  Google Scholar 

  8. Christodoulopoulou, K.: Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras. J. Algebra 320(7), 2871–2890 (2008)

    Article  MathSciNet  Google Scholar 

  9. Drozd, Y., Guzner, B., Ovsienko, S.: Weight modules over generalized Weyl algebras. J. Algebra 184(2), 491–504 (1996)

    Article  MathSciNet  Google Scholar 

  10. Futorny, V., Hartwig, J.: Multiparameter twisted Weyl algebras. J. Algebra 357, 69–93 (2012)

    Article  MathSciNet  Google Scholar 

  11. Ji, Q., Wang, D., Zhou, X.: Finite-dimensional representation of quantum group Uq(f(K)). East-West J. Math. 2(2), 201–213 (2000)

    MathSciNet  Google Scholar 

  12. Jordan, D.: Primitivity in skew Laurent polynomial rings and related rings. Math. Z. 213(3), 353–371 (1993)

    Article  MathSciNet  Google Scholar 

  13. Kostant, B.: On Whittaker vectors and representation theory. Invent. Math. 48(2), 101–184 (1978)

    Article  MathSciNet  Google Scholar 

  14. Ondrus, M.: Whittaker modules for Uq(sl2). J. Algebra 289, 192–213 (2005)

    Article  MathSciNet  Google Scholar 

  15. Ondrus, M., Wiesner, E.: Whittaker modules for the Virasoro algebra. J. Algebra Appl. 8, 363–377 (2009)

    Article  MathSciNet  Google Scholar 

  16. Rosenberg, A.: Noncommutative Algebraic Geometry and Representations of Quantized Algebras. Mathematics and Its Applications, vol. 330. Kluwer Academic Publishers Group, Dordrecht (1995)

    Google Scholar 

  17. Rueda, S.: Some algebras similar to the enveloping algebra of sl(2). Comm. Algebra 30(3), 1127–1152 (2002)

    MathSciNet  Google Scholar 

  18. Sevostyanov, A.: Quantum deformation of Whittaker modules and the Toda lattice. Duke Math. J. 105(2), 211–238 (2000)

    Article  MathSciNet  Google Scholar 

  19. Smith, S.: A class of algebras similar to the enveloping algebra of sl(2). Trans. Amer. Math. Soc. 322(1), 285–314 (1990)

    MathSciNet  Google Scholar 

  20. Tang, X.: On Whittaker modules over a class of algebras similar to U(sl2). Front. Math. China 2(1), 127–142 (2007)

    Article  MathSciNet  Google Scholar 

  21. Tang, X.: Construct irreducible representations of quantum groups Uq(fm(K)). Front. Math. China 3(3), 371–397 (2008)

    Article  MathSciNet  Google Scholar 

  22. Tang, X., Xu, Y.: On representations of quantum groups Uq(fm(K,H)). Bull. Aust. Math. Soc. 78(2), 261–284 (2008)

    Article  MathSciNet  Google Scholar 

  23. Wang, B.: Whittaker modules for graded Lie algebras. Algebra Represent. Theory 14, 691–702 (2011)

    Article  MathSciNet  Google Scholar 

  24. Wang, D., Ji, Q., Yang, S.: Finite-dimensional representations of quantum group Uq(f(K,H)). Comm. Algebra 30(5), 2191–2211 (2002)

    Article  MathSciNet  Google Scholar 

  25. Wang, L.: Non-weight modules over some generalized Weyl algebras. Thesis (Ph.D.)-The University of Science and Technology of China 67 pp (2021)

  26. Zhang, X., Tan, S., Lian, H.: Whittaker modules for the Schrödinger-Witt algebra. J. Math. Phys. 51(8), 083524, 17 (2010)

    Article  Google Scholar 

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Acknowledgements

This paper was partially supported by the NSF of China (11931009, 12161141001, 12171132 and 11771410) and Anhui Initiative in Quantum Information Technologies (AHY150000). We would also like to thank the referee for helpful comments and suggestions which improved the exposition of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

    Hongjia Chen & Longhui Wang

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  1. Hongjia Chen
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  2. Longhui Wang
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Correspondence to Longhui Wang.

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Chen, H., Wang, L. Whittaker Modules Over Some Generalized Weyl Algebras. Algebr Represent Theor 26, 3047–3064 (2023). https://doi.org/10.1007/s10468-023-10200-6

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