- Source: Radical of a Lie algebra
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- Radical of a Lie algebra
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- Levi decomposition
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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.
Definition
Let
k
{\displaystyle k}
be a field and let
g
{\displaystyle {\mathfrak {g}}}
be a finite-dimensional Lie algebra over
k
{\displaystyle k}
. There exists a unique maximal solvable ideal, called the radical, for the following reason.
Firstly let
a
{\displaystyle {\mathfrak {a}}}
and
b
{\displaystyle {\mathfrak {b}}}
be two solvable ideals of
g
{\displaystyle {\mathfrak {g}}}
. Then
a
+
b
{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}
is again an ideal of
g
{\displaystyle {\mathfrak {g}}}
, and it is solvable because it is an extension of
(
a
+
b
)
/
a
≃
b
/
(
a
∩
b
)
{\displaystyle ({\mathfrak {a}}+{\mathfrak {b}})/{\mathfrak {a}}\simeq {\mathfrak {b}}/({\mathfrak {a}}\cap {\mathfrak {b}})}
by
a
{\displaystyle {\mathfrak {a}}}
. Now consider the sum of all the solvable ideals of
g
{\displaystyle {\mathfrak {g}}}
. It is nonempty since
{
0
}
{\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
0
{\displaystyle 0}
.
A Lie algebra is reductive if and only if its radical equals its center.
See also
Levi decomposition