Hamiltonian structure, (anti-)self-adjoint flows in the KP hierarchy and the W1 + ∞ and W∞ algebras
The extended conformal WN algebras are known to be related to the generalized KdV hierarchies through their second hamiltonian structure. In this letter we discuss the relationship between the large-N limits of the WN algebras and the KP hierarchy which contains all generalized KdV hierarchies. We show that the Poisson bracket corresponding to the hamiltonian structure found by Watanabe for the KP hierarchy is isomorphic to the classical (or centerless) W1+∞ algebra, and it contains a subalgebra which is isomorphic to the W∞ algebra. Moreover, the usual generators of W1 + ∞ and W∞ are explicitly expressed in terms of the KP currents, and are shown to relate in a simple way to certain KP flows satisfying a sort of (anti-)self-duality. Our results not only clarify the underlying algebraic structure of the KP hierarchy, but also hint about a possible relationship between the latter and 4D self-dual Yang-Mills equations or gravity.