In trying to learn about $U(N)$ gauge Yang-Mills theory, I've found that what the adjoint representation actually is...is very a slippery concept for me. So I've tried to brute force generate an adjoint representation....here is what I have:
$U(2)$ is the set of all $2 \times 2$ unitary matrices. It is easy to see that the following matrices form a basis for the Lie Algebra of $U(2)$ (aka its infinitesimal generators): $$ \left(\ T^{1}, \ T^{2}, \ T^{3}, \ T^{4} \ \right) = \left(\ \frac{1}{2} \sigma_{1}, \ \frac{1}{2} \sigma_{2}, \ \frac{1}{2} \sigma_{3}, \ \frac{1}{2} \mathbb{I} \ \right) $$
...where $\sigma_{j}$ are the Pauli matrices. ${\ \bf NOTE:}$ I am using the ${\bf physicist's\ convention}$, such that $\left[ T^{a}, T^{b} \right] = i f_{abc} T^{c}$, where $f_{abc}$ are my structure constants.
I'm trying to play around with the adjoint representation ($R_{\mathrm{Adj}}$) of the above. In particular, I want to look at what the generators of the adjoint representation are (in reference to the $T^{a}$ above).
These are the matrices $T^{a}_{\mathrm{Adj}} = R_{\mathrm{Adj}} (T^{a})$. There will be $4$ of these, which are of size $4 \times 4$. It is a well-known result that these matrices have the elements determined by: \begin{eqnarray*} \left[ T^{a}_{\mathrm{Adj}} \right]_{bc} = - i f^{a}_{\ bc} \end{eqnarray*}
The game I'm trying to play is to write out these matrices explicitly. Something is happening which is freaking me out.
Since $[\sigma_{j}, \sigma_{k}] = 2 i \epsilon_{jk\ell}\sigma_{\ell}$, this means that $f_{abc} = \frac{1}{2} \epsilon_{abc}$ for $(a,b,c) \in \{1, 2, 3\}^{3}$. Also, I ${\bf think}$ that any structure constants involving the index $4$ should be zero since we're just talking about the identity. By my calculations, this means that the only non-zero structure functions are: $$ f_{123} = \tfrac{1}{2}, \ f_{132} = - \tfrac{1}{2}, \ f_{213} = - \tfrac{1}{2}, \ f_{231} = \tfrac{1}{2}, \ f_{312} = \tfrac{1}{2}, \ f_{321} = - \tfrac{1}{2}, \ $$
Which tells me that the four generators of the adjoint representation are given by: $$ \left(\ T^{1}_{\mathrm{Adj}}, \ T^{2}_{\mathrm{Adj}}, \ T^{3}_{\mathrm{Adj}}, \ T^{4}_{\mathrm{Adj}} \ \right) = \left(\ \frac{i}{2} \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right], \ \frac{i}{2} \left[ \begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right], \ \frac{i}{2} \left[ \begin{matrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right], \ \mathbb{O} \ \right) \\ $$
I am super alarmed that the fourth generator in the adjoint representation is zero....did I do something wrong? Or is it okay that it is the zero matrix? Something is not sitting right here.