From the Omega-infinity-algebra To its Central Extension - a Tau-function Approach

The KP hierarchy, deformations of pseudo-differential operators L of order one, admits a w(infinity)-algebra of symmetries Y(z)alpha(partial derivative/partial derivative(z))beta, which are vector fields transversal to and commuting with the KP hierarchy. Expressed in terms of L and another pseudo-differential operator M (introduced by Orlov and coworkers) satisfying [L,M] = 1, these vector fields act on the wave function PSI (a properly normalized eigenfunction of L) as Y(z)alpha(partial derivative/partial derivative(z))beta PSI = -(M(beta) L(alpha))-PSI. Introducing a generating function Y(N)PSI = N-PSI, with N = (mu - lambda) exp[(mu - lambda)M]delta(lambda, L), for the algebra of symmetries w(infinity) on PSI and taking into account the well-known representation of PSI(t,z) = [e-eta tau(tBAR)/tau(tBAR)] exp(SIGMA1infinity tBAR(i)z(i)), in terms of the tau-function, when eta = SIGMA(i=1)infinity(z-i/i)(partial derivative/partial derivative t(i)). We show a precise relationship between Y(N) and the Date-Jimbo-Kashiwara-Miwa vertex operator X(t, lambda, mu) = exp[SIGMA1infinity (mu(i) - lambda(i))t(i)] exp[SIGMA1infinity (lambda-i-mu-i) (1/i)(partial derivative/partial derivative t(i))], a generating function of the W(infinity)-algebra of symmetries (with central extension) on tau, to wit Y(N) log PSI = (e-eta-1)X log tau, where Y(N) log and X log act on PSI and tau as logarithmic derivatives, with respect to the vector fields Y(N) and X