Geodesic
THE GROUP OF AUTOMORPHISMS OF THE FIRST WEYL ALGEBRA IN PRIME CHARACTERISTIC AND THE RESTRICTION MAP
Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 263-274

Voir la notice de l'article provenant de la source Cambridge University Press

Let K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=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:
DOI : 10.1017/S0017089508004680
Mots-clés : 14J50, 16W20, 14L17, 14R10, 14R15, 14M20 
BAVULA, V. V. THE GROUP OF AUTOMORPHISMS OF THE FIRST WEYL ALGEBRA IN PRIME CHARACTERISTIC AND THE RESTRICTION MAP. Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 263-274. doi: 10.1017/S0017089508004680
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[1] 1.Adjamagbo, K. and van den Essen, A., On equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic, arXiv:math. AG/0608009. Google Scholar

[2] 2.Belov-Kanel, A. and Kontsevich, M., The Jacobian conjecture is stably equivalent to the Dixmier Conjecture, Mosc. Math. J. 7 (2) (2007), 209–218, arXiv:math. RA/0512171. Google Scholar | DOI

[3] 3.Belov-Kanel, A. and Kontsevich, M., Automorphisms of the Weyl algebra, Lett. Math. Phys. 74 (2) (2005), 181–199, arXiv:math. RA/0512169. Google Scholar | DOI

[4] 4.Dixmier, J., Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209–242. Google Scholar | DOI

[5] 5.Jung, H. W. E., Uber ganze birationale Transformationen der Eben, J. Reiner Angew. Math. 184 (1942), 161–174. Google Scholar | DOI

[6] 6.Humphreys, J. E., Linear Algebraic Groups, (Springer-Verlag, 1975). Google Scholar | DOI

[7] 7.Van der Kulk, W., On polynomial ring in two variables, Niew. Arch. Wisk. 1 (1953), 33–41. Google Scholar

[8] 8.Makar-Limanov, L., On automorphisms of Weyl algebra, Bull. Soc. Math. France 112 (1984), 359–363. Google Scholar | DOI

[9] 9.Kambayashi, T., Some results on pro-affine algebras and ind-affine schemes, Osaka J. Math. 40 (2003), 621–638. Google Scholar

[10] 10.Shafarevich, I. R., On some infinite-dimensional groups, Rend. Mat. Appl. 25 (1966), 208–212. Google Scholar

[11] 11.Shafarevich, I. R., On some infinite-dimensional groups-II, Math. USSR-Izvestija 18 (1982), 185–194. Google Scholar | DOI

[12] 12.Tsuchimoto, Y., Endomorphisms of Weyl algebra and p-curvatures. Osaka J. Math. 42 (2) (2005), 435–452. Google Scholar

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