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]

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.

See also

• Levi decomposition

References

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