In the mathematical field of Lie theory, the radical of a Lie algebra
is the largest solvable ideal of
[1] The radical, denoted by
{\displaystyle {\rm {rad}}({\mathfrak {g}})}
, fits into the exact sequence where
{\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}
is semisimple.
When the ground field has characteristic zero and
has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of
that is isomorphic to the semisimple quotient
{\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}
via the restriction of the quotient map
A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.
be a field and let
be a finite-dimensional Lie algebra over
There exists a unique maximal solvable ideal, called the radical, for the following reason.
be two solvable ideals of
, and it is solvable because it is an extension of
Now consider the sum of all the solvable ideals of
is a solvable ideal, and it is a solvable ideal by the sum property just derived.
Clearly it is the unique maximal solvable ideal.