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Noetherian down-up algebras

Authors: Ellen Kirkman, Ian M. Musson and D. S. Passman
Journal: Proc. Amer. Math. Soc. 127 (1999), 3161-3167
MSC (1991): Primary 16E70, 16P40
Published electronically: May 4, 1999
MathSciNet review: 1610796
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Abstract: Down-up algebras $A= A(\alpha ,\beta ,\gamma )$ were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that $\beta \neq 0$ is equivalent to $A$ being right (or left) Noetherian, and also to $A$ being a domain. Furthermore, when this occurs, we show that $A$ is Auslander-regular and has global dimension 3.

References [Enhancements On Off] (What's this?)

  • [Bv1] V. Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Canadian Math. Soc. Conf. Proc. 14 (1993), 83-107. CMP 94:09
  • [Bv2] -, Global dimension of generalized Weyl algebras, Canadian Math. Soc. Conf. Proc. 18 (1996), 81-107. MR 97e:16018
  • [B] G. Benkart, Down-up algebras and Witten's deformations of the universal enveloping algebra of $\text{sl}_{2}$, Contemporary Math. AMS (to appear).
  • [BR] G. Benkart and T. Roby, Down-up algebras, J. Algebra (to appear).
  • [Bj] J.-E. Björk, Filtered Noetherian rings, Noetherian Rings and their Applications, Math. Surveys and Monographs, vol. 24, Amer. Math. Soc., Providence, 1987, pp. 59-97.MR 89c:16018
  • [GZ] A. Giaquinto and J. Zhang, Quantum Weyl algebras, J. Algebra 176 (1995), 861-881.MR 96m:16053
  • [GW] K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative Noetherian Rings, LMS Student Text 16, Cambridge Univ. Press, Cambridge, 1989.MR 91c:16001
  • [K] Rajesh S. Kulkarni, Personal communication, 1998.
  • [L] T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow Math. J. 34 (1992), 277-300.MR 93k:16045
  • [McR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, Chichester, 1987.MR 89j:16023
  • [Z] Kaiming Zhao, Centers of down-up algebras (to appear).

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Additional Information

Ellen Kirkman
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109

Ian M. Musson
Affiliation: Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706

Received by editor(s): January 28, 1998
Published electronically: May 4, 1999
Additional Notes: This research was supported in part by NSF Grants DMS-9500486 and DMS-9622566.
Communicated by: Lance W. Small
Article copyright: © Copyright 1999 American Mathematical Society

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