In the mathematical field of Lie theory, the radical of a Lie algebra
g
is the largest solvable ideal of
g
.
Let
k
be a field and let
g
be a finite-dimensional Lie algebra over
k
. There exists a unique maximal solvable ideal, called the radical, for the following reason.
Firstly let
a
and
b
be two solvable ideals of
g
. Then
a
+
b
is again an ideal of
g
, and it is solvable because it is an extension of
(
a
+
b
)
/
a
≃
b
/
(
a
∩
b
)
by
a
. Now consider the sum of all the solvable ideals of
g
. It is nonempty since
{
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.
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.
