Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T00:35:27.205Z Has data issue: false hasContentIssue false
  • English
  • Français

A wild automorphism of Usl(2)

Published online by Cambridge University Press:  24 October 2008

A. Joseph
A. Joseph
Affiliation:
University of Tel-Aviv, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that the automorphism groups of various torsion-free associative algebras over two generators take a particularly simple form. It has been suggested (10), though never proved, that this fails over three or more generators. Here it is shown that this is indeed the case for the enveloping algebra of sl(2), a result which answers a question implicitly raised in (4). A weaker hypothesis is proposed for such automorphism groups and this is related to the structure of locally nilpotent derivations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 1 , July 1976 , pp. 61 - 65
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cohn, P. M.Free associative algebras. Bull. London Math. Soc. 1 (1969), 1–39.CrossRefGoogle Scholar
(2)Czerniakiewicz, A. J.Automorphisms of a free associative algebra of rank 2. I, II. Trans. Amer. Math. Soc. 160 (1971), 393–401; 171 (1972), 309–315.Google Scholar
(3)Dixmier, J.Sur les algèbres de Weyl. Bull. Math. Soc. France 96 (1968), 209–242.Google Scholar
(4)Dixmier, J.Quotients simples de l'algèbre enveloppante de sl(2). J. Algebra 24 (1973), 551–564.CrossRefGoogle Scholar
(5)Dixmier, J.Algèbres enveloppantes. Cahiers Scientifiques, Fascicule 37 (Paris, Gauthier-Villars, 1974).Google Scholar
(6)Jacobson, N.Lie algebras. Interscience Tracts in Pure and Applied Mathematics, 10 (New York, Interscience, 1966).Google Scholar
(7)Jung, H. W. E.Uber ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184 (1942), 161–174.CrossRefGoogle Scholar
(8)Van, Der Kulk W.On polynomial rings in two variables. Nieuw Arch. Wisk. (3) 1 (1953), 33–41.Google Scholar
(9)Makar-Limanov, L. G.Automorphisms of a free algebra with two generators. Funksional'-nyi Analiz i Ego Priloz. 4 (1970), 107–108. [English Trans. Functional Analysis and its applications 4(1970), 262–263.]Google Scholar
(10)Nagata, M. On the automorphism group of k[x, y]. Lectures in Mathematics, Kyoto University (Tokyo, Kinokuniya, 1972).Google Scholar