  A determination of the single W Spin Density Matrix (SDM) elements in the
reaction e+e- -> W+W- -> l nu q qbar (l=e/mu) is reported at centre-of-mass
energies between 189 and 209 GeV. The data sample used corresponds to an
integrated luminosity of 520 pb^{-1} taken by DELPHI between 1998 and 2000. The
single W SDM elements, rho_{tau tau'}^{W+-} (tau,tau' = +/-1 or 0), are
determined as a function of the W- production angle with respect to the e- beam
direction and are obtained from measurements of the W decay products by the
application of suitable projection operators, Lambda_{tau tau'}, which assume
the V-A coupling of the W boson to fermions. The measured SDM elements are used
to obtain the fraction of longitudinally polarised Ws, with the result:
sigma_L/sigma_tot = 24.9 +/- 4.5(stat) +/- 2.2(syst) % at a mean energy of 198
GeV. The SDM elements are also used to determine the Triple Gauge Couplings
Delta g_1^Z, Delta kappa_gamma, lambda_gamma and g_4^Z, kappa_Z and lambda_Z.
For the CP-violating couplings the results of single parameter fits are: g_4^Z
= -0.39 +0.19 -0.20 kappa_Z = -0.09 +0.08 -0.05 lambda_Z = -0.08 +/- 0.07 . The
errors are a combination of statistical and systematic errors. All results are
consistent with the Standard Model.
