Comments on the Deformed W_N Algebra

Abstract

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