Nonlinearly deformed Ŵ∞ algebra and second hamiltonian structure of KP hierarchy
The characteristic nonlinearity of WN algebras, appropriate for their many applications in two-dimensional quantum physics, is lost in the usual large-N limits. In this paper we search for nonlinear extensions of the Virasoro algebra that incorporate all higher-spin currents with spin s≥2. We show that under certain natural homogeneity requirements, the Jacobi identities lead to a unique nonlinear, centerless deformation of classical w∞ and W∞. The latter, which we call Å´∞, constitutes a universal W-algebbra which is very likely to contain all WN algebras by reduction. Also it is closely related to the linear W1+∞ by a set of interesting recursion relations, which suggests the isomorphism of Å´∞ to the second hamiltonian structure of the KP hierarchy proposed by Dickey. The implications for the symmetries in two-dimensional quantum gravity and noncritical c ≤ 1 stings in the context of the KP approach are discussed.