Lie algebras MOC

Radical of a Lie algebra

The radical rad⁡𝔤 of a Lie algebra 𝔤 is the unique maximal solvable ideal of 𝔤. 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 rad⁡𝔤 =0, which is sometimes taken as a definition.¹

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §3.1, p. 11 ↩