Abstract
As we have seen in Chapter 5, Levi’s Theorem 5.6.6 is a central result in the structure theory of Lie algebras. It often allows splitting problems: one separately considers solvable and semisimple Lie algebras, and one puts together the results for both types. Naturally, this strategy also works to some extent for Lie groups. After dealing with nilpotent and solvable Lie groups in Chapter 11, we turn to the other side of the spectrum, to groups with semisimple or reductive Lie algebras. Here an important subclass is the class of compact Lie groups and the slightly larger class of groups with compact Lie algebra. Many problems can be reduced to compact Lie groups, and they are much easier to deal with than noncompact ones. The prime reason for that is the existence of a finite Haar measure whose existence was shown in Section 10.4.
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Notes
- 1.
Note that with this definition a simple Lie group need not be simple as a group (consider, e.g., \(\mathop {\rm SL}\nolimits _{2}({\mathbb{R}})\) which has nontrivial center).
- 2.
One can show that \(G \cong \mathop {\rm Pin}\nolimits _{2}({\mathbb{R}})\) is the pin group in dimension 2, sitting in the Clifford algebra C 2≅ℍ (cf. Definition B.3.20 and Example B.3.24, where i corresponds to I∈ℍ).
- 3.
One can find this group F as a subgroup of the 16-dimensional Clifford algebra C 4 (cf. Definition B.3.4).
References
Borel, A., R. Friedman, and J. W. Morgan, Almost commuting elements in compact Lie groups, Mem. Amer. Math. Soc. 157 (2002), no. 747, x + 136pp
Bourbaki, N., Groupes et algèbres de Lie, Chapitres 9, Masson, Paris, 1982
Hofmann, K. H., and S. A. Morris, “The Structure of Compact Groups,” 2nd ed., Studies in Math., de Gruyter, Berlin, 2006
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Hilgert, J., Neeb, KH. (2012). Compact Lie Groups. In: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84794-8_12
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DOI: https://doi.org/10.1007/978-0-387-84794-8_12
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