Radical Of A Lie Algebra

In the mathematical field of Lie theory, the radical of a Lie algebra $\mathfrak$ is the largest solvable ideal of $\mathfrak.$.

The radical, denoted by $(\mathfrak)$, fits into the exact sequence
:$0 \to (\mathfrak) \to \mathfrak g \to \mathfrak/(\mathfrak) \to 0$.
where $\mathfrak/(\mathfrak)$ is semisimple. When the ground field has characteristic zero and $\mathfrak g$ has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of $\mathfrak g$ that is isomorphic to the semisimple quotient $\mathfrak/(\mathfrak)$ via the restriction of the quotient map $\mathfrak g \to \mathfrak/(\mathfrak).$

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

Definition

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

Firstly let $\mathfrak$ and $\mathfrak$ be two solvable ideals of $\mathfrak$. Then $\mathfrak+\mathfrak$ is again an ideal of $\mathfrak$, and it is solvable because it is an extension of $(\mathfrak+\mathfrak)/\mathfrak\simeq\mathfrak/(\mathfrak\cap\mathfrak)$ by $\mathfrak$. Now consider the sum of all the solvable ideals of $\mathfrak$. It is nonempty since $\$ 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 $0$.
* A Lie algebra is reductive if and only if its radical equals its center.

See also

*Levi decomposition

References

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Lie algebras