# Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the largest solvable ideal of ${\displaystyle {\mathfrak {g}}.}$[1]

## Definition

Let ${\displaystyle k}$ be a field and let ${\displaystyle {\mathfrak {g}}}$ be a finite-dimensional Lie algebra over ${\displaystyle k}$. There exists a unique maximal solvable ideal, called the radical, for the following reason.

Firstly let ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ be two solvable ideals of ${\displaystyle {\mathfrak {g}}}$. Then ${\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}$ is again an ideal of ${\displaystyle {\mathfrak {g}}}$, and it is solvable because it is an extension of ${\displaystyle ({\mathfrak {a}}+{\mathfrak {b}})/{\mathfrak {a}}\simeq {\mathfrak {b}}/({\mathfrak {a}}\cap {\mathfrak {b}})}$ by ${\displaystyle {\mathfrak {a}}}$. Now consider the sum of all the solvable ideals of ${\displaystyle {\mathfrak {g}}}$. It is nonempty since ${\displaystyle \{0\}}$ is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal.

## Related concepts

• A Lie algebra is semisimple if and only if its radical is ${\displaystyle 0}$.
• A Lie algebra is reductive if and only if its radical equals its center.

## References

1. ^ Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, Rings and Modules: Lie Algebras and Hopf Algebras, Mathematical Surveys and Monographs, 168, Providence, RI: American Mathematical Society, p. 15, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822.