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

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