[[Lie algebras MOC]]
# Radical of a Lie algebra

The **radical** $\opn{rad} \mathfrak{g}$ of a [[Lie algebra]] $\mathfrak{g}$ is the unique maximal [[Solvable Lie algebra|solvable]] [[ideal]] of $\mathfrak{g}$. #m/def/lie 

> [!check]- Proof of uniqueness
> Let $\mathfrak{r} \trianglelefteq \mathfrak{g}$ be a maximal solvable ideal, and let $\mathfrak{a} \trianglelefteq \mathfrak{g}$ be a solvable ideal.
> Then by [[Solvable Lie algebra#^P3]], $\mathfrak{r} + \mathfrak{a}$ is solvable and thus $\mathfrak{r} + \mathfrak{a} = \mathfrak{r}$ by maximality,
> whence $\mathfrak{a} \trianglelefteq \mathfrak{r}$. <span class="QED"/>

## Properties

- $\mathfrak{g}$ is a [[semisimple Lie algebra]] iff $\opn{rad} \mathfrak{g} = 0$, which is sometimes taken as a definition.[^1972]

  [^1972]: 1972\. [[Sources/@humphreysIntroductionLieAlgebras1972|Introduction to Lie Algebras and Representation Theory]], §3.1, p. 11

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