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

  • V. V. BAVULA (a1)
Abstract
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:

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

3.A. Belov-Kanel and M. Kontsevich , Automorphisms of the Weyl algebra, Lett. Math. Phys. 74 (2) (2005), 181199, arXiv:math. RA/0512169.

6.J. E. Humphreys , Linear Algebraic Groups, (Springer-Verlag, 1975).

11.I. R. Shafarevich , On some infinite-dimensional groups-II, Math. USSR-Izvestija 18 (1982), 185194.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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