Abstract
In this chapter, we shall show that for each isomorphism class of irreducible root systems there is a unique simple Lie algebra over C (up to isomorphism) with that root system. Moreover, we shall prove that every simple Lie algebra has an irreducible root system, so every simple Lie algebra arises in this way. These results mean that the classification of irreducible root systems in Chapter 13 gives us a complete classification of all complex simple Lie algebras.
We have already shown in Proposition 12.4 that if the root system of a Lie algebra is irreducible, then the Lie algebra is simple. We now show that the converse holds; that is, the root system of a simple Lie algebra is irreducible. We need the following lemma concerning reducible root systems.
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© 2006 Springer-Verlag London Limited
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Erdmann, K., Wildon, M.J. (2006). Simple Lie Algebras. In: Introduction to Lie Algebras. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/1-84628-490-2_14
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DOI: https://doi.org/10.1007/1-84628-490-2_14
Publisher Name: Springer, London
Print ISBN: 978-1-84628-040-5
Online ISBN: 978-1-84628-490-8
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