Glasgow Mathematical Journal

Research Article

THE GROUP OF AUTOMORPHISMS OF THE FIRST WEYL ALGEBRA IN PRIME CHARACTERISTIC AND THE RESTRICTION MAP

V. V. BAVULAa1

a1 Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom e-mail: v.bavula@sheffield.ac.uk

Abstract

Let K be a perfect field of characteristic p > 0; A1 := Kx, ∂|∂xx∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ 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 (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

\[
\bigg(\frac{d}{dx}+f\bigg)^p=
\bigg(\frac{d}{dx}\bigg)^p+\frac{d^{p-1}f}{dx^{p-1}}+f^p, \quad f\in K[x].
\]

(Received May 19 2008)

(Accepted September 24 2008)

2000 Mathematics Subject Classification

  • 14J50;
  • 16W20;
  • 14L17;
  • 14R10;
  • 14R15;
  • 14M20