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]

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

${\displaystyle 0\to {\rm {rad))({\mathfrak {g)))\to {\mathfrak {g))\to {\mathfrak {g))/{\rm {rad))({\mathfrak {g)))\to 0}$.

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

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

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.