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Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

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Abstract

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs.

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References

  1. Almulhem M, Brzeziński T. Skew derivations on generalized Weyl algebras. J Algebra, 2018, 493: 194–235

    Article  MathSciNet  MATH  Google Scholar 

  2. Andruskiewitsch N, Angiono I, Heckenberger I. On Nichols algebras of infinite rank with finite Gelfand-Kirillov dimension. Atti Accad Naz Lincei Rend Lincei Mat Appl, 2020, 31: 81–101

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai Y X, Chen Y Q, Zhang Z R. Gelfand-Kirillov dimension of bicommutative algebras. Linear Multilinear Algebra, 2022, in press

  4. Bao Y H, Ye Y, Zhang J J. Truncation of unitary operads. Adv Math, 2020, 372: 107290

    Article  MathSciNet  MATH  Google Scholar 

  5. Bass H. A non-triangular action of \({\mathbb{G}_a}\) on \({\mathbb{A}^3}\). J Pure Appl Algebra, 1984, 33: 1–5

    Article  MathSciNet  MATH  Google Scholar 

  6. Bavula V V. Generalized Weyl algebras and their representations. St Petersburg Math J, 1993, 4: 71–92

    MathSciNet  MATH  Google Scholar 

  7. Bavula V V. Filter dimension of algebras and modules, a simplicity criterion of generalized Weyl algebras. Comm Algebra, 1996, 24: 1971–1992

    Article  MathSciNet  MATH  Google Scholar 

  8. Bavula V V. Global dimension of generalized Weyl algebras. In: Representation Theory of Algebras, vol. 18. Providence: Amer Math Soc, 1996, 81–107

    MATH  Google Scholar 

  9. Bavula V V, Jordan D A. Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans Amer Math Soc, 2001, 353: 769–794

    Article  MathSciNet  MATH  Google Scholar 

  10. Bavula V V, Lenagan T H. Krull dimension of generalized Weyl algebras with noncommutative coefficients. J Algebra, 2001, 235: 315–358

    Article  MathSciNet  MATH  Google Scholar 

  11. Bavula V V, Lu T. The quantum Euclidean algebra and its prime spectrum. Israel J Math, 2017, 219: 929–958

    Article  MathSciNet  MATH  Google Scholar 

  12. Bavula V V, van Oystaeyen F. Krull dimension of generalized Weyl algebras and iterated skew polynomial rings: Commutative coefficients. J Algebra, 1998, 208: 1–34

    Article  MathSciNet  MATH  Google Scholar 

  13. Bell J P, Zelmanov E. On the growth of algebras, semigroups, and hereditary languages. Invent Math, 2021, 224: 683–697

    Article  MathSciNet  MATH  Google Scholar 

  14. Bell J P, Zhang J J. Zariski cancellation problem for noncommutative algebras. Selecta Math (NS), 2017, 23: 1709–1737

    Article  MathSciNet  MATH  Google Scholar 

  15. Bruns W, Gubeladze J. Polytopes, Rings, and K-Theory. New York: Springer, 2009

    MATH  Google Scholar 

  16. Brzeziński T. Noncommutative differential geometry of generalized Weyl algebras. SIGMA Symmetry Integrability Geom Methods Appl, 2016, 12: 059

    MathSciNet  MATH  Google Scholar 

  17. Ceken S, Palmieri J H, Wang Y H, et al. The discriminant controls automorphism groups of noncommutative algebras. Adv Math, 2015, 269: 551–584

    Article  MathSciNet  MATH  Google Scholar 

  18. Chan K, Gaddis J, Won R, et al. Reflexive hull discriminants and applications. Selecta Math (NS), 2022, 28: 40

    Article  MathSciNet  MATH  Google Scholar 

  19. Ferraro L, Gaddis J, Won R. Simple ℤ-graded domains of Gelfand-Kirillov dimension two. J Algebra, 2020, 562: 433–465

    Article  MathSciNet  MATH  Google Scholar 

  20. Furter J-P. Quasi-locally finite polynomial endomorphisms. Math Z, 2009, 263: 473–479

    Article  MathSciNet  MATH  Google Scholar 

  21. Gaddis J, Won R. Fixed rings of generalized Weyl algebras. J Algebra, 2019, 536: 149–169

    Article  MathSciNet  MATH  Google Scholar 

  22. Gelfand I M, Kirillov A A. Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. Publ Math Inst Hautes Etudes Sci, 1966, 31: 5–19

    Article  MATH  Google Scholar 

  23. Goodearl K R, Zhang J J. Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two. Glasg Math J, 2017, 59: 563–593

    Article  MathSciNet  MATH  Google Scholar 

  24. Grigorchuk R, Pak I. Groups of intermediate growth: An introduction. Enseign Math, 2008, 54: 251–272

    MathSciNet  MATH  Google Scholar 

  25. Gutiérrez J, Valqui C. Bivariant K-theory of generalized Weyl algebras. J Noncommut Geom, 2020, 14: 639–666

    Article  MathSciNet  MATH  Google Scholar 

  26. Huh C, Kim C O. Gelfand-Kirillov dimension of skew polynomial rings of automorphism type. Comm Algebra, 1996, 24: 2317–2323

    Article  MathSciNet  MATH  Google Scholar 

  27. Jordan D A, Sasom N. Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms. J Algebra Appl, 2009, 8: 733–757

    Article  MathSciNet  MATH  Google Scholar 

  28. Jung H W E. Über ganze birationale transformationen der Ebene. J Reine Angew Math, 1942, 184: 161–174

    Article  MathSciNet  MATH  Google Scholar 

  29. Khoroshkin A, Piontkovski D. On generating series of finitely presented operads. J Algebra, 2015, 426: 377–429

    Article  MathSciNet  MATH  Google Scholar 

  30. Klyuev D. Twisted traces and positive forms on generalized q-Weyl algebras. SIGMA Symmetry Integrability Geom Methods Appl, 2022, 18: 009

    MathSciNet  MATH  Google Scholar 

  31. Krause G R, Lenagan T H. Growth of Algebras and Gelfand-Kirillov Dimension. Providence: Amer Math Soc, 2000

    MATH  Google Scholar 

  32. Lane D R. Fixed points of affine Cremona transformations of the plane over an algebraically closed field. Amer J Math, 1975, 97: 707–732

    Article  MathSciNet  MATH  Google Scholar 

  33. Leroy A, Matczuk J, Okninski J. On the Gelfand-Kirillov dimension of normal localizations and twisted polynomial rings. In: Perspectives in Ring Theory. NATO ASI Series, vol. 233. Dordrecht: Springer, 1988, 205–214

    MATH  Google Scholar 

  34. Levitt G, Nicolas J-L. On the maximum order of torsion elements in GL(n, Z) and Aut(F n). J Algebra, 1998, 208: 630–642

    Article  MathSciNet  MATH  Google Scholar 

  35. Lezama O, Wang Y H, Zhang J J. Zariski cancellation problem for non-domain noncommutative algebras. Math Z, 2019, 292: 1269–1290

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu L Y. On homological smoothness of generalized Weyl algebras over polynomial algebras in two variables. J Algebra, 2018, 498: 228–253

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu L Y, Ma W. Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras. Front Math China, 2022, in press

  38. Lorenz M. On the Gelfand-Kirillov dimension of skew polynomial rings. J Algebra, 1982, 77: 186–188

    Article  MathSciNet  MATH  Google Scholar 

  39. Matczuk J. The Gelfand-Kirillov dimension of Poincare-Birkhoff-Witt extensions. In: Perspectives in Ring Theory. NATO ASI Series, vol. 233. Dordrecht: Springer, 1988, 221–226

    MATH  Google Scholar 

  40. McConnell J C, Robson J C. Noncommutative Noetherian Rings. Graduate Studies in Mathematics, vol. 30. Providence: Amer Math Soc, 2001

    MATH  Google Scholar 

  41. Milnor J. A note on curvature and fundamental group. J Differential Geom, 1968, 2: 1–7

    Article  MathSciNet  MATH  Google Scholar 

  42. Nagata M. On Automorphism Group of k[x, y]. Tokyo: Kinokuniya, 1972

    MATH  Google Scholar 

  43. Qi Z H, Xu Y J, Zhang J J, Zhao X G. Growth of nonsymmetric operads. Indiana Univ Math J, 2022, in press

  44. Rogalski D. GK-dimension of birationally commutative surfaces. Trans Amer Math Soc, 2009, 361: 5921–5945

    Article  MathSciNet  MATH  Google Scholar 

  45. Shestakov I P, Umirbaev U U. Poisson brackets and two-generated subalgebras of rings of polynomials. J Amer Math Soc, 2003, 17: 181–196

    Article  MathSciNet  MATH  Google Scholar 

  46. Smith M K. Universal enveloping algebras with subexponential but not polynomially bounded growth. Proc Amer Math Soc, 1976, 60: 22–24

    Article  MathSciNet  MATH  Google Scholar 

  47. Suárez-Alvarez M, Vivas Q. Automorphisms and isomorphisms of quantum generalized Weyl algebras. J Algebra, 2015, 424: 540–552

    Article  MathSciNet  MATH  Google Scholar 

  48. Tang X, Zhang J J, Zhao X G. Cancellation of Morita and skew types. Israel J Math, 2021, 244: 467–500

    Article  MathSciNet  MATH  Google Scholar 

  49. van der Kulk W. On polynomial rings in two variables. Nieuw Arch Wiskd (5), 1953, 3: 33–41

    MathSciNet  MATH  Google Scholar 

  50. Wang D G, Zhang J J, Zhuang G. Connected Hopf algebras of Gelfand-Kirillov dimension four. Trans Amer Math Soc, 2015, 367: 5597–5632

    Article  MathSciNet  MATH  Google Scholar 

  51. Won R. The noncommutative schemes of generalized Weyl algebras. J Algebra, 2018, 506: 322–349

    Article  MathSciNet  MATH  Google Scholar 

  52. Wu Q S. Gelfand-Kirillov dimension under base field extension. Israel J Math, 1991, 73: 289–296

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang J J. A note on GK dimension of skew polynomial extensions. Proc Amer Math Soc, 1997, 125: 363–373

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhang Y, Zhao X G. Gelfand-Kirillov dimension of differential difference algebras. LMS J Comput Math, 2014, 17: 485–495

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhao X G, Mo Q H, Zhang Y. Gelfand-Kirillov dimension of generalized Weyl algebras. Comm Algebra, 2018, 46: 4403–4413

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by Huizhou University (Grant Nos. hzu202001 and 2021JB022) and the Guangdong Provincial Department of Education (Grant Nos. 2020KTSCX145 and 2021ZDJS080). The author thanks the referees for their careful reading, helpful suggestions and useful references. Part of this work was done during a visit to James J. Zhang at the University of Washington. The author expresses deep gratitude to James J. Zhang for many stimulating conversations on the subject, and to the department of Mathematics at the University of Washington for the hospitality during his visit. The author wishes to thank Yu Li, Wenchao Zhang and Xiaotong Sun for helpful discussions.

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  1. School of Mathematics and Statistics, Huizhou University, Huizhou, 516007, China

    Xiangui Zhao

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  1. Xiangui Zhao
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Correspondence to Xiangui Zhao.

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Dedicated to Professor Yuqun Chen on the Occasion of His 65th Birthday

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Zhao, X. Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras. Sci. China Math. 66, 887–906 (2023). https://doi.org/10.1007/s11425-022-1992-2

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