6 - Complex semisimple Lie algebras

Summary

In this chapter, we begin the study of semisimple Lie algebras and their representations. This is one of the highest achievements of the theory of Lie algebras, which has numerous applications (for example, to physics), not to mention that it is also one of the most beautiful areas of mathematics.

Throughout this chapter, g is a finite-dimensional semisimple Lie algebra (see Definition 5.35); unless specified otherwise, g is complex.

Properties of semisimple Lie algebras

Cartan's criterion of semimplicity, proved in Section 5.8, is not very convenient for practical computations. However, it is extremely useful for theoretical considerations.

Proposition 6.1.Let g be a real Lie algebra and g_ℂ – its complexification (see Definition 3.49). Then g is semisimple iff g_ℂis semisimple.

Proof. Immediately follows from Cartan's criterion of semisimplicity.

Remark 6.2. This theorem fails if we replace the word “semisimple” by “simple”: there exist simple real Lie algebras g such that g_ℂ is a direct sum of two simple algebras.

Theorem 6.3.Let g be a semisimple Lie algebra, and I ⊂ g – an ideal. Then there is an ideal I' such that g = I ⊕ I'.

Proof. Let I^⊥ be the orthogonal complement with respect to the Killing form. By Lemma 5.45, I^⊥ is an ideal. Consider the intersection I ∩ I^⊥. It is an ideal in g with zero Killing form (by Exercise 5.1). Thus, by Cartan criterion, it is solvable.