Cartan’s Criteria

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 EquationSource$$ tr\left( {\left[ {a,b} \right]c} \right) = tr\left( {a\left[ {b,c} \right]} \right) $$ 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.KeywordsComplex Vector SpaceKilling FormSimple IdealJordan DecompositionJordan Normal FormThese 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.