In the mathematical field of Lie theory, the radical of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is the largest solvable ideal of
g
.
{\displaystyle {\mathfrak {g}}.}
The radical, denoted by
r
a
d
(
g
)
{\displaystyle {\rm {rad}}({\mathfrak {g}})}
, fits into the exact sequence
0
→
r
a
d
(
g
)
→
g
→
g
/
r
a
d
(
g
)
→
0
{\displaystyle 0\to {\rm {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})\to 0}
.
where
g
/
r
a
d
(
g
)
{\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}
is semisimple. When the ground field has characteristic zero and
g
{\displaystyle {\mathfrak {g}}}
has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of
g
{\displaystyle {\mathfrak {g}}}
that is isomorphic to the semisimple quotient
g
/
r
a
d
(
g
)
{\displaystyle {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})}
via the restriction of the quotient map
g
→
g
/
r
a
d
(
g
)
.
{\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.
