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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra in prime characteristic


Author: V. V. Bavula
Journal: Proc. Amer. Math. Soc. 137 (2009), 1891-1898
MSC (2000): Primary 16W20, 13N10, 16S32
Published electronically: January 26, 2009
MathSciNet review: 2480268
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Abstract: Let $ K$ be a field of characteristic $ p>0$. It is proved that the group $ \mathrm{Aut}_{ord}(\mathcal{D}(L_n))$ of order preserving automorphisms of the ring $ \mathcal{D}(L_n)$ of differential operators on a Laurent polynomial algebra $ L_n:= K[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$ is isomorphic to a skew direct product of groups $ {\mathbb{Z}}_p^n \rtimes \mathrm{Aut}_K(L_n)$, where $ {\mathbb{Z}}_p$ is the ring of $ p$-adic integers. Moreover, the group $ \mathrm{Aut}_{ord}(\mathcal{D}(L_n))$ is found explicitly. Similarly, $ \mathrm{Aut}_{ord}(\mathcal{D}(P_n))\simeq \mathrm{Aut}_K(P_n)$, where $ P_n: =K[x_1, \ldots, x_n]$ is a polynomial algebra.


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Additional Information

V. V. Bavula
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: v.bavula@sheffield.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09825-6
PII: S 0002-9939(09)09825-6
Keywords: Group of automorphisms, ring of differential operators, the order filtration
Received by editor(s): June 4, 2008
Published electronically: January 26, 2009
Communicated by: Martin Lorenz
Article copyright: © Copyright 2009 American Mathematical Society