Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$? I just watched this lecture and there Susskind says that 
$${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$
for the Lie group $G= SU(N)$. Unfortunately, he does not offer any explanation for this. Does anyone know some good explanation?

We have $\mathfrak{su}(n)\otimes\mathbb{C}\cong\mathfrak{sl}(n,\mathbb{C})$ acting on $N=\mathbb{C}^n$. Since $N\otimes N^\ast\cong\mathrm{End}(N)$ as representations, this is the representation of $\mathfrak{sl}(n,\mathbb{C})$ on $\mathfrak{gl}(n,\mathbb{C})$, which internally decomposes as
$$ \mathfrak{gl}(n,\mathbb{C})=\mathbb{C}\cdot\mathrm{Id}\oplus\mathfrak{sl}(n,\mathbb{C}). $$
