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I'm trying to show that the generators of the adjoint representation of $U(N)$ are hermitian. I know that; $$ U(N) \ = \ \left\{ \ U \in \mathbb{M}_{N \times N}(\mathbb{C})\ | \ U^{\dagger} = U^{-1} \ \right\} \\ \mathfrak{u}(N) \ = \ \left\{ \ T \in \mathbb{M}_{N \times N}(\mathbb{C})\ | \ T^{\dagger} = -T\ \right\} $$ Bear with me because I've been taught what the adjoint representation is by physicists (lol).

I'm told it's the representation such that $\mathrm{Ad}_{g}(T_{a}) = g T^{a} g^{-1}$, where $g \in U(N)$ and $T^{a}$ are the generators of $\mathfrak{u}(N)$. Supposedly we treat the Lie algebra $\mathfrak{u}(N)$ as the vector space over which the Lie Group acts.

My question is, how do I even determine WHAT the generators of the adjoint representation are? I have no clue how to start doing this question. (Surely the generators of the adjoint representation are not just $T^{a}$? This seems too simple.)

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