An extended integrable model of the KdV hierarchy and the resulting Hamiltonian structure

An algebraic system and its monomial system are constructed, on which an algebraic operation relation is given. It follows that a new expression of the zero curvature equation is manifested, which is used to generate the KdV hierarchy. Furthermore, an extended Liouville integrable model of the famous KdV hierarchy is obtained, whose Hamiltonian structure is worked out with the help of the quadratic-form identity.