Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-09T17:17:06.433Z Has data issue: false hasContentIssue false
  • English
  • Français

A family of simple groups associated with the Satake diagrams

Part of: Other groups of matrices

Published online by Cambridge University Press:  09 April 2009

Cheng Chonhu
Cheng Chonhu
Affiliation:
Department of Mathematics, Xiangtan University, Xiangtanm, Hunan Province, Peoples' Republic of China
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.

Using the theory of the Satake diagrams associated with the non-compact simple Lie algebras over the real number field R, we shall construct a family of simple groups over a field K which are called the simple groups associated with the Satake diagrams. The list of these simple groups includes all Chevalley groups and twisted groups, and all simple algebraic groups of adjoint type defined over R if K is the complex number field C (except two types given by Table II′). Furthermore, the simple groups associated with the Satake diagrams of type AIII, BI, DI are identified with the simple groups obtained from the unitary or orthogonal groups of non-zero indices. The quasi-Bruhat decomposition of the “non-split” simple groups associated with the Satake diagrams which are not Chevalley groups or twisted groups will be given in this paper.

MSC classification

Secondary: 20H20: Other matrix groups over fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 13 - 43
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Carter, R. W., Simple groups of Lie type, (Wiley, London, New York, 1972).Google Scholar
[2]ChonHu, Cheng, ‘Some simple groups of Lie type constructed by the inner automorphism’, Chinese Ann. Math. 1 (1980), 161–176.Google Scholar
[3]Chevalley, C., ‘Sur certain groupes simples’, Tohoku Math. J. 7 (1955), 14–66.CrossRefGoogle Scholar
[4]Helgason, S., Differential geometry, Lie groups, and symmetric spaces (Academic Press, New York, 1978).Google Scholar
[5]Satake, I., Classification theory of semisimple algebraic groups’, (Dekker, New York, 1971).Google Scholar
[6]Steinberg, R., ‘Variations on a theme of Chevalley’, Pacific J. Math. 9 (1959), 875–891.CrossRefGoogle Scholar
[7]Tits, J., ‘Classification of algebraic semisimple groups’, pp. 32–62 Proceedings of Symposia in Pure Math. Vol. 9, Algebraic groups and discontinous subgroups.Google Scholar