$W_n^{(\text{ka})}$ algebra associated with the Moyal KdV Hierarchy

We consider the Gelfand-Dickey (GD) structure defined by the Moyal $\star$-product with parameter $\text{ka}$, which not only defines the bi-Hamiltonian structure for the generalized Moyal KdV hierarchy but also provides a $W_n^{(\text{ka})}$ algebra containing the Virasoro algebra as a subalgebra with central charge ${\text{ka}}^2(n^3-n)\mathrm{/}3$. The free-field realization of the $W_n^{(\text{ka})}$ algebra is given through the Miura transformation and the cases for $W_3^{(\text{ka})}$ and $W_4^{(\text{ka})}$ are worked out in detail.