Abstract
How can one decide whether a complex Lie algebra is semisimple? Working straight from the definition, one would have to test every single ideal for solvability, seemingly a daunting task. In this chapter, we describe a practical way to decide whether a Lie algebra is semisimple or, at the other extreme, solvable, by looking at the traces of linear maps.
We have already seen examples of the usefulness of taking traces. For example, we made an essential use of the trace map when proving the Invariance Lemma (Lemma 5.5). An important identity satisfied by trace is
for linear transformations a, b, c of a vector space. This holds because tr b(ac) = tr(ac)b; we shall see its usefulness in the course of this chapter. Furthermore, note that a nilpotent linear transformation has trace zero.
From now on, we work entirely over the complex numbers.
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© 2006 Springer-Verlag London Limited
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Erdmann, K., Wildon, M.J. (2006). Cartan’s Criteria. In: Introduction to Lie Algebras. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/1-84628-490-2_9
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DOI: https://doi.org/10.1007/1-84628-490-2_9
Publisher Name: Springer, London
Print ISBN: 978-1-84628-040-5
Online ISBN: 978-1-84628-490-8
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