ON THE GELFAND-DICKEY ALGEBRA GD(SLN) AND THE WN SYMMETRY .1. THE BOSONIC CASE/

The algebra XI of nonlinear (local and nonlocal) differential operator s, acting on the ring of analytic functions, is studied. It is shown i n particular that this space splits into 3X2 special subalgebras SIGMA (jr), j=0,+/-1, r=+/-1. Each subalgebra is completely specified by qua ntum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra SIGMA++ (and its d ual SIGMA--) of local (pure nonlocal) differential operators is used t o calculate the general expression of the Gelfand-Dickey bracket and t he W(n)-symmetry Poisson, one in terms of a set of spin j canonical fi elds u(j), 2 less-than-or-equal-to j less-than-or-equal-to n and a non linear u-cubic dependent differential operator D (n,i,j;z,u). The expl icit form of this operator is worked out. Other remarkable features ar e also discussed.