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The group of automorphisms of the first weyl algebra in prime characteristic and the restriction map

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

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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].

Item Type: Article
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.
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: 08 Feb 2013 16:58
Published Version: http://dx.doi.org/10.1017/S0017089508004680
Status: Published
Publisher: Cambridge University Press
Refereed: Yes
Identification Number: 10.1017/S0017089508004680
URI: http://eprints.whiterose.ac.uk/id/eprint/8651

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