Radical of a Lie algebra

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

The radical, denoted by [math]\displaystyle{ {\rm rad}(\mathfrak{g}) }[/math], fits into the exact sequence

[math]\displaystyle{ 0 \to {\rm rad}(\mathfrak{g}) \to \mathfrak g \to \mathfrak{g}/{\rm rad}(\mathfrak{g}) \to 0 }[/math].

where [math]\displaystyle{ \mathfrak{g}/{\rm rad}(\mathfrak{g}) }[/math] is semisimple. When the ground field has characteristic zero and [math]\displaystyle{ \mathfrak g }[/math] has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of [math]\displaystyle{ \mathfrak g }[/math] that is isomorphic to the semisimple quotient [math]\displaystyle{ \mathfrak{g}/{\rm rad}(\mathfrak{g}) }[/math] via the restriction of the quotient map [math]\displaystyle{ \mathfrak g \to \mathfrak{g}/{\rm rad}(\mathfrak{g}). }[/math]

A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.

Definition

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

Firstly let [math]\displaystyle{ \mathfrak{a} }[/math] and [math]\displaystyle{ \mathfrak{b} }[/math] be two solvable ideals of [math]\displaystyle{ \mathfrak{g} }[/math]. Then [math]\displaystyle{ \mathfrak{a}+\mathfrak{b} }[/math] is again an ideal of [math]\displaystyle{ \mathfrak{g} }[/math], and it is solvable because it is an extension of [math]\displaystyle{ (\mathfrak{a}+\mathfrak{b})/\mathfrak{a}\simeq\mathfrak{b}/(\mathfrak{a}\cap\mathfrak{b}) }[/math] by [math]\displaystyle{ \mathfrak{a} }[/math]. Now consider the sum of all the solvable ideals of [math]\displaystyle{ \mathfrak{g} }[/math]. It is nonempty since [math]\displaystyle{ \{0\} }[/math] 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 [math]\displaystyle{ 0 }[/math].
• A Lie algebra is reductive if and only if its radical equals its center.

See also

• Levi decomposition