Polarized top decay into polarized W: t(up)->W(up) + b at O(alpha_s)

We consider the decay of a polarized top quark into a polarized W -boson plus a bottom quark followed by the decay of the W -boson into a pair of leptons or quarks. The polar angle distribution of the top spin relative to the W -momentum and the polar angle distribution of the lepton (or quark) in the W -rest frame is governed by three polarized and three unpolarized rate functions which are related to the double density matrix elements of the decay $t \to W^+ + b$. We obtain analytical expressions for the $O(\alpha_s)$ radiative corrections to the three polarized and three unpolarized rate functions. We also provide a comprehensive discussion of the dependence of the longitudinal and transverse polarization of the top quarks produced at e^+e^--colliders on beam polarization parameters.