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

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

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

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

(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 $L$ ,

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

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

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

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,

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

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