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examples

EXAMPLE 5.45   Let $ k$ be a field of characteristic $ p$ (possibly 0 ).

$\displaystyle \mathfrak{gl}_n(k)
$

is a Lie algebra with the Killing form

    $\displaystyle \kappa_{\mathfrak{gl}_n(k)}(x,y)=$ $\displaystyle \operatorname{tr}((\lambda(x)-\rho(x))(\lambda(y)-\rho(y)))$
    $\displaystyle =$ $\displaystyle \operatorname{tr}(\lambda(x y)) +\operatorname{tr}(\rho(x y)) -\operatorname{tr}(\lambda(x)\rho(y)-\operatorname{tr}(\lambda(y)\rho(x))$
    $\displaystyle =$ $\displaystyle 2n \operatorname{tr}(x y)-2 \operatorname{tr}(x)\operatorname{tr}(y).$

$ \mathfrak{sl}_n(k)$ is an ideal of $ \mathfrak{gl}_n(k)$ and so its Killing form is given by

$\displaystyle \kappa_{\mathfrak{sl}_n(k)}(x,y)=2 n\operatorname{tr}_{k^n}(x y).
$

If $ p \not \vert 2 n$ , then the Killing form is easily seen to be non-degenerate. so $ \mathfrak{sl}_n(k)$ is a non-degenerated Lie algebra in this case. In this way we see that it is a semisimple Lie algebra. The Lie algebra is actually simple as we have shown in Proposition 5.22.

EXAMPLE 5.46   Let $ p$ be an odd prime. Let $ k$ be a field of characteristic $ p$ . Then we have shown in Proposition 5.19 that the only non trivial ideals of $ \mathfrak{gl}_p(k)$ are $ \mathfrak{sl}_p(k)$ and $ k.1_p$ . So we see that

$\displaystyle L=\mathfrak{gl}_p(k)/k. 1_p
$

is a semisimple Lie algebra (as it has no proper abelian ideals). It has a unique nontrivial ideal

$\displaystyle M=\mathfrak{sl}_p(k)/k. 1_p.
$

Thus $ L$ cannot be a direct sum of simple Lie algebras.


next up previous
Next: Weyl's theorem on complete Up: generalities in finite dimensional Previous: Cartan's criterion for semisimplicity
2007-12-19