Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:17:19.171Z Has data issue: false hasContentIssue false
  • English
  • Français

On Closed Subsets of Root Systems

Published online by Cambridge University Press:  20 November 2018

D. Ž. Doković ,
P. Check  and
J.-Y. Hée
D. Ž. Doković
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1
P. Check
Affiliation:
Department of Mathematics University of Michigan Ann Arbor, Michigan 48109-1003 U.S.A.
J.-Y. Hée
Affiliation:
Université Paris-Sud Mathématique, Bât. 425 91405 Orsay, Cedex France
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.

Let R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

Keywords

17B67
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994 , pp. 338 - 345
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Bourbaki, N., Groupes etalgèbres de Lie, Chapitres 4, 5, et 6, Hermann, Paris, 1968.Google Scholar
2. Carter, R. W., Simple groups of Lie type, Wiley, J. W., New York, 1972.Google Scholar
3. Futorny, V. M., The parabolic subsets of root system and corresponding representations of affine Lie algebras, Contemporary Math. 131(1992), 45–52.Google Scholar
4. Malyshev, F. M., Decompositions of root systems, Math. Notes Acad. Sci. USSR 27(1980), 418–421.Google Scholar
5. Warner, G., Harmonic analysis on semi-simple Lie groups I, Springer, New York, 1972.Google Scholar