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