MATH 223A NOTES 2011
LIE ALGEBRAS
17
5.
Semisimple Lie algebras and the Killing form
This section follows Procesi’s book on Lie Groups.
We will define semisimple Lie
algebras and the Killing form and prove the following.
Theorem 5.0.9.
The following are equivalent for
L
a finite dimensional Lie algebra over
any subfield
F
⊆
C
.
(1)
L
is semisimple.
(2)
L
has no nonzero abelian ideals.
(3)
The Killing form of
L
is nondegenerate.
(4)
L
is a direct sum of simple ideals.
5.1.
Definition.
First we observe that the sum of two solvable ideals
I, J
in
L
is solvable.
This follows from the fact that
I
and (
I
+
J
)
/I
=
J/
(
I
∩
J
) are both solvable.
Definition 5.1.1.
The
solvable radical
Rad
L
of
L
is defined to be the sum of all solvable
ideals. A Lie algebra is
semisimple
if its solvable radical is zero, i.e., if it has no nonzero
solvable ideal.
Proposition 5.1.2.
L
is semisimple i
ff
L
has no nonzero abelian ideals.
Proof.
If
L
is semisimple then it has no abelian ideals. Conversely, if
L
is not semisimple,
then
L
has a solvable ideal
J
. Then
DJ
= [
JJ
] is also an ideal in
L
since
[
x
[
JJ
]]
⊆
[[
xJ
]
J
] + [
J
[
xJ
]]
⊆
[
JJ
]
We have
D
k
J
= 0. So
D
k
−
1
J
is a nonzero abelian ideal in
L
.
5.2.
Killing form.
The
Killing form
κ
:
L
×
L
→
F
is defined by
κ
(
x, y
) = Tr(ad
x
ad
y
)
The Killing form is clearly symmetric:
κ
(
x, y
) =
κ
(
y, x
). The Killing form is also “asso
ciative”:
κ
([
xy
]
, z
) =
κ
(
x,
[
yz
])
Proof.
Since ad[
xy
] = [ad
x,
ad
y
], we have:
κ
([
xy
]
, z
) = Tr(ad [
xy
] ad
z
) = Tr([ad
x,
ad
y
]ad
z
) = Tr(ad
x
[ad
y,
ad
z
]) =
κ
(
x,
[
yz
])
Proposition 5.2.1.
The Killing form is invariant under any automorphism
ρ
of
L
.
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 Fall '11
 K.Igusa
 Algebra, Lie algebra, Killing form, Semisimple Lie algebra, Procesi

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