Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-30T22:34:45.512Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Fundamental Group of a Simple Lie Group

Published online by Cambridge University Press:  22 January 2016

Masaru Takeuchi
Masaru Takeuchi*
Affiliation:
Osaka University
Rights & Permissions [Opens in a new window]

Extract

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 G be a simply connected simple Lie group and C the center of G, which is isomorphic with the fundamental group of the adjoint group of G. For an element c of C, an element x of the Lie algebra g of G is called a representative of c in g if exp x = c. Sirota-Solodovnikov [7] found a complete set of representatives of the center C in g and studied the group structure of C, and using their results Goto-Kobayashi [1] classified subgroups of the center C with respect to automorphisms of G. The group structure of C was also studied in Takeuchi [8],

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 40 , December 1970 , pp. 147 - 159
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1970

References

[1] Goto, M. and Kobayashi, E.: On the subgroups of the centers of simply connected simple Lie groups-classification of simple Lie groups in the large, to appear.Google Scholar
[2] Helgason, S.: Differential geometry and symmetric spaces, New York, 1962.Google Scholar
[3] Iwahori, N. and Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. des Hautes Etudes Sci. Pub. Math. 25(1965), 237–280.Google Scholar
[4] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329–387.Google Scholar
[5] Murakami, S.: On the automorphisms of a real semi-simple Lie algebra, J. Math. Soc. Japan 4 (1952), 103–133.Google Scholar
[6] Murakami, S.: Sur la classification des algebres de Lie reelles et simples, Osaka J. Math. 2 (1965), 291–307.Google Scholar
[7] Sirota, A.I. and Solodovnikov, A.S.: Non-compact semi-simple Lie groups, Uspekhi Mat. Nauk 18, No. 3 (1963), 87–144. English translation of the same issue, 85–140.Google Scholar
[8] Takeuchi, M.: On the fundamental group and the group of isometries of a symmetric space, J. Fac. Sci. Univ. Tokyo, I, 10(1964), 88–123.Google Scholar