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Transactions of the American Mathematical Society

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$ \mathbb{Z}$-graded simple rings


Authors: J. Bell and D. Rogalski
Journal: Trans. Amer. Math. Soc. 368 (2016), 4461-4496
MSC (2010): Primary 16D30, 16P90, 16S38, 16W50
DOI: https://doi.org/10.1090/tran/6472
Published electronically: October 14, 2015
MathSciNet review: 3453377
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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
Email: jpbell@uwaterloo.ca

D. Rogalski
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
Email: drogalsk@math.ucsd.edu

DOI: https://doi.org/10.1090/tran/6472
Keywords: Noncommutative geometry, graded ring, generalized Weyl algebra, simple ring
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.
Article copyright: © Copyright 2015 American Mathematical Society

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