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. 263274. ISSN 00170895
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 (nonobvious) equality proved in the paper: (d/dx + f)(p) = (d/dx)(p) + d(p1)f/dx(p1) + f(p), f is an element of K[x].
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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 selfarchiving policy. 
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:  https://doi.org/10.1017/S0017089508004680 
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