\(W_n^{(\kappa)}\) algebra associated with the Moyal KdV hierarchy

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Abstract: We consider the Gelfand-Dickey (GD) structure defined by the Moyal

s t a r

-product with parameter

k a

, which not only defines the bi-Hamiltonian structure for the generalized Moyal KdV hierarchy but also provides a

W_{n}^{(k a)}

algebra containing the Virasoro algebra as a subalgebra with central charge

k a^{2} (n^{3} - n) / 3

. The free-field realization of the

W_{n}^{(k a)}

algebra is given through the Miura transformation and the cases for

W_{3}^{(k a)}

and

W_{4}^{(k a)}

are worked out in detail.

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