  The non-linear W_{\infty}[\mu] symmetry algebra underlies the duality between
the W_N minimal model CFTs and the hs[\mu] higher spin theory on AdS_3. It is
shown how the structure of this symmetry algebra at the quantum level, i.e. for
finite central charge, can be determined completely. The resulting algebra
exhibits an exact equivalence (a`triality') between three (generically)
distinct values of the parameter \mu. This explains, among other things, the
agreement of symmetries between the W_N minimal models and the bulk higher spin
theory. We also study the consequences of this triality for some of the
simplest W_{\infty}[\mu] representations, thereby clarifying the analytic
continuation between the`light states' of the minimal models and conical defect
solutions in the bulk. These considerations also lead us to propose that one of
the two scalar fields in the bulk actually has a non-perturbative origin.
