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functoriality of Killing forms

LEMMA 5.27   Let $ L$ be a Lie algebra over a field $ k$ . Then the followings are true.
  1. Let $ V$ be a finite dimensional representation of $ L$ . Let $ W$ be a subrepresentation of $ V$ . Then we have

    $\displaystyle \operatorname{Tr}_V(x y)= \operatorname{Tr}_W (x y)+\operatorname{Tr}_{V/W} (x y).
$

  2. Let $ I$ be an ideal of $ L$ . Assume $ L$ is finite dimensional. Then we have

    $\displaystyle \kappa_L(x, y)=\operatorname{Tr}_{\operatorname{ad}, L}(x y)
=\op...
...)
=\operatorname{Tr}_{\operatorname{ad},I}(x y)+\kappa_{L/I}(\bar{x}, \bar{y})
$

    (where $ \bar{\bullet}$ denotes the class of $ \bullet $ in $ L/I$ .) In particular, for any $ x,y\in I$ , we have

    $\displaystyle \kappa_L(x, y)= \kappa_I(x, y)
$

PROOF.. (2): We choose a basis $ B=B_1\coprod B_2$ of $ L$ such that $ B_2$ forms a basis of $ I$ . Then $ \bar{B_1}$ forms a basis of $ L/I$ . Under the basis $ B$ , $ \operatorname{ad}(x)$ may be represented by a matrix

$\displaystyle \operatorname{ad}_L (x)=
\begin{pmatrix}
\operatorname{ad}_{L/I}(\bar{x}) & * \\
0 & \operatorname{ad}_I{x}
\end{pmatrix}.
$

We obtain the result easily from this.

(1): may be proved in a same manner. $ \qedsymbol$


next up previous
Next: Theorem of Iwasawa Up: Invariant bilinear forms and Previous: Invariant bilinear forms and
2009-03-06