The group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra in prime characteristic

Author:
V. V. Bavula

Journal:
Proc. Amer. Math. Soc. **137** (2009), 1891-1898

MSC (2000):
Primary 16W20, 13N10, 16S32

DOI:
https://doi.org/10.1090/S0002-9939-09-09825-6

Published electronically:
January 26, 2009

MathSciNet review:
2480268

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a field of characteristic . It is proved that the group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra is isomorphic to a skew direct product of groups , where is the ring of -adic integers. Moreover, the group is found explicitly. Similarly, , where is a polynomial algebra.

**1.**H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse,*Bull. Amer. Math. Soc. (New Series)*,**7**(1982), 287-330. MR**663785 (83k:14028)****2.**V. V. Bavula, The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic,*J. Pure Appl. Algebra*,**212**(2008), 2320-2337. MR**2426512****3.**V. V. Bavula, Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic, arXiv:math.RA/0804.1091.**4.**V. V. Bavula, The implies the , arXiv:math. RA/0512250.**5.**V. V. Bavula, The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map,*Glasgow Math. J.*, to appear (arXiv:math.RA/0708.1620).**6.**A. Belov-Kanel and M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture,*Mosc. Math. J.*,**7**(2007), no. 2, 209-218. MR**2337879****7.**J. Dixmier, Sur les algèbres de Weyl.*Bull. Soc. Math. France*,**96**(1968), 209-242. MR**0242897 (39:4224)****8.**H. W. E. Jung, Über ganze birationale Transformationen der Ebene,*J. Reine Angew. Math.*,**184**(1942), 161-174. MR**0008915 (5:74f)****9.**Y. Tsuchimoto, Endomorphisms of Weyl algebra and -curvatures.*Osaka J. Math.*,**42**(2005), no. 2, 435-452. MR**2147727 (2006g:14101)****10.**W. van der Kulk, On polynomial rings in two variables,*Nieuw. Arch. Wisk.*(3) 1 (1953), 33-41. MR**0054574 (14:941f)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16W20,
13N10,
16S32

Retrieve articles in all journals with MSC (2000): 16W20, 13N10, 16S32

Additional Information

**V. V. Bavula**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Email:
v.bavula@sheffield.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-09-09825-6

Keywords:
Group of automorphisms,
ring of differential operators,
the order filtration

Received by editor(s):
June 4, 2008

Published electronically:
January 26, 2009

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2009
American Mathematical Society