  We introduce a Frobenius algebra-valued KP hierarchy and show the existence
of Frobenius algebra-valued $\tau$-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 P.Casati and G.Ortenzi in \cite{CO2006} 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.
