Bavula, V.V. (2009) The group of automorphisms of the first weyl algebra in prime characteristic and the restriction map. Glasgow Mathematical Journal, 51 (2). pp. 263-274. ISSN 0017-0895
Abstract
Let K be a perfect field of characteristic p > 0; A(1) := K < x, partial derivative vertical bar partial derivative x - x partial derivative = 1 > be the first Weyl algebra; and Z := K[X := x(p), Y := partial derivative(p)] be its centre. It is proved that (1) the restriction map res : Aut(K)(A(1)) -> Aut(K)(Z), sigma bar right arrow sigma vertical bar(Z) is a monomorphism with im(res) = Gamma := (tau is an element of Aut(K)(Z) vertical bar J(tau) = 1), where J(tau) is the Jacobian of tau, (Note that Aut(K)(Z) = K* (sic) Gamma, and if K is not perfect then im(res) not equal Gamma.); (ii) the bijection res : Aut(K)(A(1)) -> Gamma is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res(-1) is found via differential operators D(Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper: (d/dx + f)(p) = (d/dx)(p) + d(p-1)f/dx(p-1) + f(p), f is an element of K[x].
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2009 Glasgow Mathematical Journal Trust . This is an author produced version of a paper subsequently published in Glasgow Mathematical Journal. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Miss Anthea Tucker |
Date Deposited: | 26 Jun 2009 08:42 |
Last Modified: | 16 Nov 2015 11:48 |
Published Version: | http://dx.doi.org/10.1017/S0017089508004680 |
Status: | Published |
Publisher: | Cambridge University Press |
Refereed: | Yes |
Identification Number: | 10.1017/S0017089508004680 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:8651 |