Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

We introduce a Frobenius algebra-valued Kadomtsev-Petviashvili (KP) hierarchy and show the existence of Frobenius algebra-valued τ-function for this hierarchy. In addition, we construct its Hamiltonian structures by using the Adler-Dickey-Gelfand method. As a byproduct of these constructions, we show that the coupled KP hierarchy, defined by Casati and Ortenzi [J. Geom. Phys. 56, 418-449 (2006)], has at least n-"basic" different local bi-Hamiltonian structures. Finally, via the construction of the second Hamiltonian structures, we obtain some local matrix, or Frobenius algebra-valued, generalizations of classical W-algebras.