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Lie Groups pp 191-201 | Cite as

The Fundamental Group

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Abstract

In this chapter, we will look more closely at the fundamental group of a compact Lie group G. We will show that it is a finitely generated Abelian group and that each loop in G can be deformed into any given maximal torus. Then we will show how to calculate the fundamental group. Along the way we will encounter another important Coxeter group, the affine Weyl group. The key arguments in this chapter are topological and are adapted from Adams [2].

Keywords

Fundamental Group Maximal Torus Bruhat Decomposition Affine Weyl Group Simple Positive Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    J. Adams. Lectures on Lie Groups. W. A. Benjamin, Inc., New York-Amsterdam, 1969.zbMATHGoogle Scholar
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    Nicolas Bourbaki. Lie groups and Lie algebras. Chapters  4 6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley.
  3. 24.
    Nicolas Bourbaki. Lie groups and Lie algebras. Chapters  7 9. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley.
  4. 86.
    N. Iwahori and H. Matsumoto. On some Bruhat decomposition and the structure of the Hecke rings of \(\mathfrak{p}\)-adic Chevalley groups. Inst. Hautes Études Sci. Publ. Math., 25:5–48, 1965.MathSciNetCrossRefzbMATHGoogle Scholar
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    E. Spanier. Algebraic Topology. McGraw-Hill Book Co., New York, 1966.zbMATHGoogle Scholar
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    E. Stiefel. Kristallographische Bestimmung der Charaktere der geschlossenen Lie’schen Gruppen. Comment. Math. Helv., 17:165–200, 1945.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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