What are the properties of central derivations in Lie algebras?

[Pivych]

I need to solve two assignments in Lie algebras. These assignments are not very difficult, but my knowledges in Lie algebras aren't good.
1. Let $$\delta$$ be a derivation of the Lie algebra $$\Im$$. Show that if $$\delta$$ commutes with every inner derivation, then $$\delta$$($$\Im$$)$$\subseteq$$C($$\Im$$), where C($$\Im$$) denotes the centre of $$\Im$$ .

2. Let x $$\in$$ gl(n,F) have n distinct eigenvalues $$\lambda$$1..$$\lambda$$n in F. Prove that eigenvalues of ad$$_{}x$$ are the n$$^{}2$$ scalars $$\lambda$$$$_{}i$$-$$\lambda$$$$_{}j$$ (1$$\leq$$i,j$$\leq$$n)

Your prompt reply will be highly appreciated

 

[henry_m]

1. Let $$\delta$$ be a derivation of the Lie algebra $$\Im$$. Show that if $$\delta$$ commutes with every inner derivation, then $$\delta$$($$\Im$$)$$\subseteq$$C($$\Im$$), where C($$\Im$$) denotes the centre of $$\Im$$ .

You need to show that for all x and y, $$\delta(x)$$ commutes with y, i.e. $$[y,\delta(x)]=0$$. What's a central derivation? Can you rewite the condition using one?

2. Let x $$\in$$ gl(n,F) have n distinct eigenvalues $$\lambda$$1..$$\lambda$$n in F. Prove that eigenvalues of ad$$_{}x$$ are the n$$^{}2$$ scalars $$\lambda$$$$_{}i$$-$$\lambda$$$$_{}j$$ (1$$\leq$$i,j$$\leq$$n)

x is an matrix, its eigenvectors are vectors in Fⁿ. The map ad_x acts on elements of the LA, i.e. matrices over F. So its eigenvectors are matrices, and you can construct them directly. Try to think of ways to construct matrices from vectors.