Killing form

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 . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle k}$ ${\displaystyle \operatorname {ad} :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}({\mathfrak {g}})}$

The killing form is the through

${\displaystyle B(X,Y):=\operatorname {Tr} (\operatorname {ad} (X)\circ \operatorname {ad} (Y))}$

for defined symmetrical bilinear form ${\displaystyle X,Y\in {\mathfrak {g}}}$

${\displaystyle B:{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow k}$,

where denotes the track . ${\displaystyle \operatorname {Tr} }$

properties

• ${\displaystyle B}$ is a symmetrical bilinear form.
• ${\displaystyle B}$is associative , that is, it applies to everyone . ${\displaystyle B([X,Y],Z)=B(X,[Y,Z])}$ ${\displaystyle X,Y,Z\in {\mathfrak {g}}}$
• For all is skew-symmetrical with respect to , that means for all applies ${\displaystyle Z\in {\mathfrak {g}}}$ ${\displaystyle \operatorname {ad} (Z)}$ ${\displaystyle B}$ ${\displaystyle X,Y\in {\mathfrak {g}}}$

${\displaystyle B(\operatorname {ad} (Z)X,Y)=-B(X,\operatorname {ad} (Z)Y)}$.

• The killing form is non-degenerate if and only if the Lie algebra is semi-simple. ${\displaystyle {\mathfrak {g}}}$
• If the Lie algebra is a Lie group , then -invariant, i. H. for all true ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle \operatorname {Ad} }$ ${\displaystyle g\in G,X,Y\in {\mathfrak {g}}}$

${\displaystyle B(\operatorname {Ad} (g)X,\operatorname {Ad} (g)Y)=B(X,Y)}$.

• 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 . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$ ${\displaystyle -B}$ ${\displaystyle G}$

Examples

The killing form of nilpotent Lie algebras is identically zero.

For many classic Lie algebras the killing form can be specified explicitly:

G	${\displaystyle B(X,Y)}$
gl ( n , R )	${\displaystyle 2n\operatorname {Tr} (XY)-2\operatorname {Tr} (X)\operatorname {Tr} (Y)}$
sl ( n , R )	${\displaystyle 2n\operatorname {Tr} (XY)}$
su ( n )	${\displaystyle 2n\operatorname {Tr} (XY)}$
so ( n , R )	${\displaystyle (n-2)\operatorname {Tr} (XY)}$
so ( n )	${\displaystyle (n-2)\operatorname {Tr} (XY)}$
sp ( n , R )	${\displaystyle (2n+2)\operatorname {Tr} (XY)}$
sp ( n , C )	${\displaystyle (2n+2)\operatorname {Tr} (XY)}$

Riemannian metric on symmetric spaces of non-compact type

A symmetric space of non-compact type is a manifold of shape

${\displaystyle M=G/K}$

with a semi-simple Lie group and a maximally compact subgroup . ${\displaystyle G}$ ${\displaystyle K}$

For a symmetrical space one has a Cartan decomposition

${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$

and one can identify the tangent space in the neutral element with . ${\displaystyle T_{\left[e\right]}G/K}$ ${\displaystyle {\mathfrak {p}}}$

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 . ${\displaystyle {\mathfrak {k}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle \operatorname {Ad} (G)}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle M=G/K}$ ${\displaystyle G}$ ${\displaystyle M}$

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 . ${\displaystyle 0}$

literature

• Humphreys, James E .: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972.