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Invariant bilinear forms and Killing forms

DEFINITION 5.24   A symmetric bilinear form $ B: L\times L \to k$ of a Lie algebra over a field $ k$ is said to be invariant if it satisfies

$\displaystyle B([Y,X] , Z)+B(X,[Y,Z])=0 \qquad(\forall X,Y,Z\in L)
$

(which means that ``the Lie derivative of $ B$ is zero''), or, equivalently,

$\displaystyle B([X,Y] , Z)=B(X,[Y,Z]) \qquad(\forall X,Y,Z\in L)
$

(which means that $ B$ is ``balanced''.)

LEMMA 5.25   Let $ L$ be a Lie algebra over a field $ k$ . Let $ B$ be an invariant bilinear form on $ L$ . Then for any ideal $ I$ of $ L$ ,

$\displaystyle L^\perp=\{x\in L; B(x,y)=0 (\forall y\in l)\}
$

is an ideal of $ L$ .

PROOF.. Easy. $ \qedsymbol$

Note: We need to be a bit careful when we use the notation $ \bullet^\perp$ . It is safer to clarify the ``container'' ($ L$ ) and bilinear form $ \rho$ . So the lemma above we should have written $ L^{\perp_{\rho,L}}$ (eek) in stead of $ L^\perp$ .

DEFINITION 5.26   Let $ (\rho,V)$ be a finite dimensional representation of a Lie algebra $ L$ over a field $ k$ . Then the Killing form with respect to $ (\rho,V)$ is a bilinear form on $ L$ defined by

$\displaystyle \operatorname{Tr}_{\rho,V}(X Y)=\operatorname{tr}_{V}(\rho(X)\rho(Y)).
$

The ordinary(usual) Killing form$ \kappa_L$ of $ L$ is a bilinear form on $ L$ defined as the Killing form of the adjoint representation. That is,

$\displaystyle \kappa_L(X,Y)=\operatorname{Tr}_{\operatorname{ad},L}(X Y)= \operatorname{tr}_{\operatorname{ad},L}(\operatorname{ad}(X)\operatorname{ad}(Y)).
$

It is easy to verify that the Killing forms defined as above are invariant.



Subsections
next up previous
Next: functoriality of Killing forms Up: generalities in finite dimensional Previous: Ideals of .
2007-12-19