Radical of a Lie algebra explained
is the largest
solvable ideal of
[1] The radical, denoted by
, fits into the exact sequence
0\to{\rmrad}(ak{g})\toakg\toak{g}/{\rmrad}(ak{g})\to0
.where
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
via the restriction of the quotient map
akg\toak{g}/{\rmrad}(ak{g}).
A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.
Definition
Let
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.
Firstly let
and
be two solvable ideals of
. Then
is again an ideal of
, and it is solvable because it is an
extension of
(ak{a}+ak{b})/ak{a}\simeqak{b}/(ak{a}\capak{b})
by
. Now consider the sum of all the solvable ideals of
. 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
.
- A Lie algebra is reductive if and only if its radical equals its center.
See also
Notes and References
- .
