Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-08T12:46:42.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Orders in simple Lie algebras of Chevalley type

Published online by Cambridge University Press:  09 April 2009

James F. Hurley
James F. Hurley
Affiliation:
Department of Mathematics Research School of Phisical Sciences The Australian National University Canberra, A.C.T.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The integral structure of a simple Lie algebra L of Chevalley type over a field F of fractions of an integral domain D is studied. Sandwich relations for sufficiently large orders are obtained, including a new general sandwich relation for orders of L in case D is an integrally closed Noetherian domain. Generalizations of the principal results of Hyman (1966) in the case when D is a ring of algebraic integers are obtained, using techniques developed by the author and Stewart (1973) which are applied to certain orders in L that arise in a natural fashion from the Chevalley basis.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 17 B 20; secondary 17 B 10, 17 B 45, 20 G 15, 20 G 05.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1978 , pp. 76 - 85
Copyright
Copyright © Australian Mathematical Society 1978

References

Abe, E. (1969), “Chevalley groups over local rings”, Tohoku J. Math. 21, 474–494.CrossRefGoogle Scholar
Abe, E. and Suzuki, K. (1976), “On normal subgroups of Chevalley groups over commutative rings”, Tohoku J. Math. 28, 185–198.CrossRefGoogle Scholar
Chevalley, C. (1955), “Sur certains groupes simples”, Tohoku J. Math. 7, 14–66.CrossRefGoogle Scholar
Chevalley, C. (1961), Séminaire Bourbaki, Fasc. 1, Exp. 219 (Secrétariat Mathématique, Paris).Google Scholar
Curtis, C. W. and Reiner, I. (1962), Representation Theory of Finite Groups and Associative Algebras(Wiley—Interscience, New York).Google Scholar
Hurley, J. F. (1969), “Ideals in Chevalley algebras”, Trans. Amer. Math. Soc. 137, 245–258.CrossRefGoogle Scholar
Hurley, J. F. (1971), “Extensions of Chevalley algebras”, Duke Math. J. 38, 349–356.CrossRefGoogle Scholar
Hurley, J. F. (1971), “Some normal subgroups of elementary subgroups of Chevalley groups over rings”, American J. Math. 93, 1059–1069.CrossRefGoogle Scholar
Hyman, M. H. (1966), “Some aspects of the arithmetic theory of Lie algebras”, Ph.D. Dissertation, Yale University, New Haven, Conn.Google Scholar
Jacobson, N. (1953), Lectures in Abstract Algebra, Vol. II (Van Nostrand, Princeton, N.J.).CrossRefGoogle Scholar
Jacobson, N. (1962), Lie Algebras (Wiley–Interscience, New York).Google Scholar
Kostant, B. (1966), “Groups over Z”, Proc. Symposia Pure Math. 9, 90–98.CrossRefGoogle Scholar
Seligman, G. B. (1967), Modular Lie Algebras (Springer-Verlag, New York, Heidelberg and Berlin).CrossRefGoogle Scholar
Springer, T. A. (1966), “Some arithmetical results on semisimple Lie algebras”, Inst. Hautes Études Sci. Publ. Math. 30, 475–501.CrossRefGoogle Scholar
Steinberg, R. (1965), “Regular elements of semisimple algebraic groups”, Inst. Hautes Études Sci. Publ. Math. 25, 49–80.CrossRefGoogle Scholar
Steinberg, R. (1968), Lectures on Chevalley Groups (mimeographed, Yale Univ. Math. Dept., New Haven, Conn.).Google Scholar
Stewart, I. (1973), “Central simplicity and Chevalley algebras”, Compositio Math. 26, 111–118.Google Scholar
Zariski, O. and Samuel, P. (1958), Commutative Algebra, Vol. I (Van Nostrand, Princeton, N.J.).Google Scholar