Radical of a Lie algebra

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

Definition

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

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