In a quantum field theory book (Schwartz, Quantum Field Theory and the Standard Model), I came across the statement that "all finite-dimensional representations of semisimple Lie algebras are hermitian".
I did not find any reference citing this result.
In addition, is it not $\mathfrak{sl}(2, \mathbb{C})$ a semisimple Lie algebra, with a finite-dimensional representation which is not Hermitian (or skew-Hermitian)?
E.g. I take the well-known representation
$$X=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, Y=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, H=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$$
Am I misunderstanding the statement? Sorry for the triviality of the question.
Thank you