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

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