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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$\mathbb {Z}$-graded simple rings
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by J. Bell and D. Rogalski PDF
Trans. Amer. Math. Soc. 368 (2016), 4461-4496 Request permission

Abstract:

The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb {Z}$-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study $\mathbb {Z}$-graded simple rings $A$ of any dimension which have a graded quotient ring of the form $K[t, t^{-1}; \sigma ]$ for a field $K$. Under some further hypotheses, we classify all such $A$ in terms of a new construction of simple rings which we introduce in this paper. In the important special case that $\operatorname {GKdim} A = \operatorname {tr.deg}(K/k) + 1$, we show that $K$ and $\sigma$ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry.
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Additional Information
  • J. Bell
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
  • MR Author ID: 632303
  • Email: jpbell@uwaterloo.ca
  • D. Rogalski
  • Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
  • MR Author ID: 734142
  • Email: drogalsk@math.ucsd.edu
  • Received by editor(s): October 29, 2013
  • Received by editor(s) in revised form: May 3, 2014
  • Published electronically: October 14, 2015
  • Additional Notes: The first author was partially supported by NSERC grant 31-611456.
    The second author was partially supported by NSF grants DMS-0900981 and DMS-1201572.
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 4461-4496
  • MSC (2010): Primary 16D30, 16P90, 16S38, 16W50
  • DOI: https://doi.org/10.1090/tran/6472
  • MathSciNet review: 3453377