Take the 2-minute tour ×  Is there any reference for the
following fact? I am looking for a nice and simple proof.
Assume that $G=GL(n,C)$, the group of invertible $n\times n$
matrices with complex entries. Why can the dual of the Lie
algebra of $G$ can be identified with the vector space of all
$n\times n$ matrices, where $G$ acts by conjugation?
share|improve this question  1 Answer 1  up vote 4 down vote
accepted  Use $\text{trace}_{\mathbb C}(X\alpha)$ as duality
pairing between $X$ in the Lie algebra and $\alpha$ in the dual;
both matrices.  Edit: More details. It is a non degenerate
duality, compatible with conjugation, since
$\text{trace}(gXg^{-1}.\alpha)=\text{trace}(X.g^{-1}\alpha g)$.
Note that $(g,\alpha)\mapsto g^{-1}\alpha g$ is the coadjoint
action.  share|improve this answer @Peter MICHOR, Do you have
any reference? –  Hassan Jolany Jun 10 '13 at 17:09 I would be
happy if you give a little bit with detail –  Hassan Jolany Jun
10 '13 at 17:11 @Hassan: What Peter is emphasizing is the
existence