A new algebraic system and its applications

In terms of the properties of the known loop algebra Ã1 and difference operators, a new algebraic system X is constructed, which is devote to working out the well-known Toda lattice hierarchy and the Schrödinger lattice hierarchy, respectively. Then an extended algebraic system X̃ of X is presented, from which the integrable coupling systems of the Toda lattice and the Schrödinger lattice are obtained.