1. Jul 25, 2014

### center o bass

In my limited study of abstract Lie groups, I have come across the adjoint representation $Ad: G \to GL(\mathfrak{g})$ on the lie algebra $\matfrak{g}$. It is defined through the conjugation map $C_g(h) = ghg^{-1}$ as the pushforward $C_{g*}|_{g=e}: \mathfrak{g} \to \mathfrak{g}$.

Does this bear any relation to that one would call the adjoint of a matrix? I.e. the operation where one transposes and takes the complex conjugate?

2. Jul 29, 2014

### WWGD

I a not sure, but I would think the answer is no, at least not in the case of groups since when A a V (finite-dim. since we are using matrices)in the target space , i.e., A is in GL(V) , then A^T is a map in End(V*) , not in GL(V).