Comments on the Deformed WN Algebra

We obtain an explicit expression for the defining relation of the deformed W_N algebra, DWA(widehat {sl}_N)_q,t. Using this expression we can show that, in the q-->1 limit, DWA(widehat {sl}_N)q,t with t=e^-(2πi)/(N)} q^{k+N}/N reduces to the _N-version of the Lepowsky-Wilson's Z-algebra of level k, ZA(widehat {sl}_{N)_k}. In other words DWA (widehat {sl}_N)q,t with t=e^{-(2π i)/(N)}q^{k+N}/N can be considered as a q-deformation of ZA (widehat {sl}_N)_k. In the appendix given by H. Awata, S. Odake and J. Shiraishi, we present an interesting relation between DWA (widehat {sl}_N)_q,t and ζ-function regularization.