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

Abstract

Let K be a perfect field of characteristic p > 0; A(1) := K 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]