1. ## Lie algebras

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 be a derivation of the Lie algebra . Show that if commutes with every inner derivation, then ()C(), where C() denotes the centre of .

2. Let x gl(n,F) have n distinct eigenvalues 1..n in F. Prove that eigenvalues of ad are the n scalars - (1i,jn)

2. Originally Posted by 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 be a derivation of the Lie algebra . Show that if commutes with every inner derivation, then ()C(), where C() denotes the centre of .

2. Let x gl(n,F) have n distinct eigenvalues 1..n in F. Prove that eigenvalues of ad are the n scalars - (1i,jn)

let $L$ be our Lie algebra. i will write $ad_a$ for inner derivations. so we are given that $\delta ad_a(b) = ad_a \delta(b)$ for all $a,b \in L$. that means $\delta[a,b]=[a,\delta(b)]$ for all $a,b \in L$. call this (1). we also have, by definition of a derivation, $\delta[a,b]=[\delta(a),b] + [a,\delta(b)]$. call this (2). now (1) and (2) gives you $[\delta(a),b]=0$ for all $a,b \in L$. that means $\delta(a)$ is central for all $a \in L$ and we're done.
for the second question use the fact that a matrix with distinct eigenvalues is diagonalizable. so suppose that $\{v_1, \ldots , v_n \}$ is a basis for $F^n$ such that the matrix of $x$ in this basis is diagonal. that means $x(v_i)=\lambda_i v_i$ for all $i$. now for any $1 \leq i,j \leq n$ define $e_{ij} \in gl(n,F)$ by $e_{ij}(v_k)=\delta_{jk}v_i$ for all $k$ where $\delta_{jk}$ is the Kronecker delta. see that these $e_{ij}$ make a basis for $gl(n,F)$ and
$ad_x e_{ij}(v_k)=(xe_{ij}-e_{ij}x)(v_k)=(\lambda_i - \lambda_k)\delta_{jk}v_i=(\lambda_i - \lambda_j)e_{ij}(v_k).$
hence $ad_xe_{ij}=(\lambda_i - \lambda_j)e_{ij}$ and so the matrix of $ad_x$ relative to the basis $\{e_{ij}, \ 1 \leq i,j \leq n \}$ is diagonal and its diagonal entries are $\lambda_i - \lambda_j$.