The killing form (also called Cartan-Killing form ) plays an important role in differential geometry and in the classification of the semi-simple Lie algebras . It is named after Wilhelm Killing .
definition
Let be a Lie algebra over the field and its adjoint representation .
G
{\ displaystyle {\ mathfrak {g}}}
k
{\ displaystyle k}
ad
:
G
→
G
l
(
G
)
{\ displaystyle \ operatorname {ad}: {\ mathfrak {g}} \ rightarrow {\ mathfrak {gl}} ({\ mathfrak {g}})}
The killing form is the through
B.
(
X
,
Y
)
: =
Tr
(
ad
(
X
)
∘
ad
(
Y
)
)
{\ displaystyle B (X, Y): = \ operatorname {Tr} (\ operatorname {ad} (X) \ circ \ operatorname {ad} (Y))}
for defined symmetrical bilinear form
X
,
Y
∈
G
{\ displaystyle X, Y \ in {\ mathfrak {g}}}
B.
:
G
×
G
→
k
{\ displaystyle B: {\ mathfrak {g}} \ times {\ mathfrak {g}} \ rightarrow k}
,
where denotes the track .
Tr
{\ displaystyle \ operatorname {Tr}}
properties
B.
{\ displaystyle B}
is a symmetrical bilinear form.
B.
{\ displaystyle B}
is associative , that is, it applies to everyone .
B.
(
[
X
,
Y
]
,
Z
)
=
B.
(
X
,
[
Y
,
Z
]
)
{\ displaystyle B ([X, Y], Z) = B (X, [Y, Z])}
X
,
Y
,
Z
∈
G
{\ displaystyle X, Y, Z \ in {\ mathfrak {g}}}
For all is skew-symmetrical with respect to , that means for all applies
Z
∈
G
{\ displaystyle Z \ in {\ mathfrak {g}}}
ad
(
Z
)
{\ displaystyle \ operatorname {ad} (Z)}
B.
{\ displaystyle B}
X
,
Y
∈
G
{\ displaystyle X, Y \ in {\ mathfrak {g}}}
B.
(
ad
(
Z
)
X
,
Y
)
=
-
B.
(
X
,
ad
(
Z
)
Y
)
{\ displaystyle B (\ operatorname {ad} (Z) X, Y) = - B (X, \ operatorname {ad} (Z) Y)}
.
The killing form is non-degenerate if and only if the Lie algebra is semi-simple.
G
{\ displaystyle {\ mathfrak {g}}}
If the Lie algebra is a Lie group , then -invariant, i. H. for all true
G
{\ displaystyle {\ mathfrak {g}}}
G
{\ displaystyle G}
B.
{\ displaystyle B}
Ad
{\ displaystyle \ operatorname {Ad}}
G
∈
G
,
X
,
Y
∈
G
{\ displaystyle g \ in G, X, Y \ in {\ mathfrak {g}}}
B.
(
Ad
(
G
)
X
,
Ad
(
G
)
Y
)
=
B.
(
X
,
Y
)
{\ displaystyle B (\ operatorname {Ad} (g) X, \ operatorname {Ad} (g) Y) = B (X, Y)}
.
If the Lie algebra is a semisimple Lie group , then the killing form is negative definite if and only if is compact. In particular, it defines a bi-invariant Riemannian metric on a compact, semi-simple Lie group . More generally, on the Lie algebra of a compact (not necessarily semi-simple) Lie group, the killing form is always negative semidefinite .
G
{\ displaystyle {\ mathfrak {g}}}
G
{\ displaystyle G}
-
B.
{\ displaystyle -B}
G
{\ displaystyle G}
Examples
The killing form of nilpotent Lie algebras is identically zero.
For many classic Lie algebras the killing form can be specified explicitly:
G
B.
(
X
,
Y
)
{\ displaystyle B (X, Y)}
gl ( n , R )
2
n
Tr
(
X
Y
)
-
2
Tr
(
X
)
Tr
(
Y
)
{\ displaystyle 2n \ operatorname {Tr} (XY) -2 \ operatorname {Tr} (X) \ operatorname {Tr} (Y)}
sl ( n , R )
2
n
Tr
(
X
Y
)
{\ displaystyle 2n \ operatorname {Tr} (XY)}
su ( n )
2
n
Tr
(
X
Y
)
{\ displaystyle 2n \ operatorname {Tr} (XY)}
so ( n , R )
(
n
-
2
)
Tr
(
X
Y
)
{\ displaystyle (n-2) \ operatorname {Tr} (XY)}
so ( n )
(
n
-
2
)
Tr
(
X
Y
)
{\ displaystyle (n-2) \ operatorname {Tr} (XY)}
sp ( n , R )
(
2
n
+
2
)
Tr
(
X
Y
)
{\ displaystyle (2n + 2) \ operatorname {Tr} (XY)}
sp ( n , C )
(
2
n
+
2
)
Tr
(
X
Y
)
{\ displaystyle (2n + 2) \ operatorname {Tr} (XY)}
Riemannian metric on symmetric spaces of non-compact type
A symmetric space of non-compact type is a manifold of shape
M.
=
G
/
K
{\ displaystyle M = G / K}
with a semi-simple Lie group and a maximally compact subgroup .
G
{\ displaystyle G}
K
{\ displaystyle K}
For a symmetrical space one has a Cartan decomposition
G
=
k
⊕
p
{\ displaystyle {\ mathfrak {g}} = {\ mathfrak {k}} \ oplus {\ mathfrak {p}}}
and one can identify the tangent space in the neutral element with .
T
[
e
]
G
/
K
{\ displaystyle T _ {\ left [e \ right]} G / K}
p
{\ displaystyle {\ mathfrak {p}}}
The killing form is negative definitely on and positive definitely on . In particular, it defines an -invariant scalar product and thus a left-invariant Riemannian metric . Except for multiplication by scalars, this is the only -invariant metric .
k
{\ displaystyle {\ mathfrak {k}}}
p
{\ displaystyle {\ mathfrak {p}}}
Ad
(
G
)
{\ displaystyle \ operatorname {Ad} (G)}
p
{\ displaystyle {\ mathfrak {p}}}
M.
=
G
/
K
{\ displaystyle M = G / K}
G
{\ displaystyle G}
M.
{\ displaystyle M}
The differential geometry of symmetrical spaces deals with the properties of these Riemannian manifolds.
Classification of semi-simple Lie algebras
The killing form plays a key role in the classification of the semi-simple Lie algebras over algebraically closed fields of the characteristic .
0
{\ displaystyle 0}
literature
Humphreys, James E .: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972.
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