Radical of a Lie algebra

The radical $r a d 𝔤$ of a Lie algebra $𝔤$ is the unique maximal solvable ideal of $𝔤$ . #m/def/lie

Proof of uniqueness

Let $𝔯 ⊴ 𝔤$ be a maximal solvable ideal, and let $𝔞 ⊴ 𝔤$ be a solvable ideal. Then by ^P3, $𝔯 + 𝔞$ is solvable and thus $𝔯 + 𝔞 = 𝔯$ by maximality, whence $𝔞 ⊴ 𝔯$ .

Properties

$𝔤$ is a semisimple Lie algebra iff $r a d 𝔤 = 0$ , which is sometimes taken as a definition.¹

#state/tidy | #lang/en | #SemBr

1972. Introduction to Lie Algebras and Representation Theory, §3.1, p. 11 ↩

Radical of a Lie algebra

Properties

Footnotes