Radical_of_a_Lie

Radical of a Lie algebra

The radical of a Lie algebra $\mathfrak{g}$ is a particular ideal of $\mathfrak{g}$ .

Definition

Let $\mathfrak{g}$ be a Lie algebra. The radical of $\mathfrak{g}$ is defined as the largest solvable ideal of $\mathfrak{g}$ .

Such an ideal exists for the following reason. 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 $\mathfrak{a}$ . Therefore we may also define the radical of $\mathfrak{g}$ as the sum of all the solvable ideals of $\mathfrak{g}$ .

Relation with semisimple Lie algebras

A Lie algebra is semisimple if its radical is $0$ .

Definition

Relation with semisimple Lie algebras

See also