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].
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References
J. Adams. Lectures on Lie Groups. W. A. Benjamin, Inc., New York-Amsterdam, 1969.
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.
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.
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.
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E. Stiefel. Kristallographische Bestimmung der Charaktere der geschlossenen Lie’schen Gruppen. Comment. Math. Helv., 17:165–200, 1945.
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Bump, D. (2013). The Fundamental Group. In: Lie Groups. Graduate Texts in Mathematics, vol 225. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8024-2_23
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DOI: https://doi.org/10.1007/978-1-4614-8024-2_23
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