Coset Vertex Operator Algebras and $\W$-Algebras

We give an explicit description for the weight three generator of the coset vertex operator algebra $C_{L_{\widehat{\sl_{n}}}(l,0)\otimes L_{\widehat{\sl_{n}}}(1,0)}(L_{\widehat{\sl_{n}}}(l+1,0))$, for $n\geq 2, l\geq 1$. Furthermore, we prove that the commutant $C_{L_{\widehat{\sl_{3}}}(l,0)\otimes L_{\widehat{\sl_{3}}}(1,0)}(L_{\widehat{\sl_{3}}}(l+1,0))$ is isomorphic to the $\W$-algebra $\W_{-3+\frac{l+3}{l+4}}(\sl_3)$, which confirms the conjecture for the $\sl_3$ case that $C_{L_{\widehat{\frak g}}(l,0)\otimes L_{\widehat{\frak g}}(1,0)}(L_{\widehat{\frak g}}(l+1,0))$ is isomorphic to $\W_{-h+\frac{l+h}{l+h+1}}(\frak g)$ for simply-laced Lie algebras ${\frak g}$ with its Coxeter number $h$ for a positive integer $l$.