The quadratic-form identity for constructing the Hamiltonian structures of the discrete integrable systems

The quadratic-form identity is extended to the discrete version which can be used to construct the Hamiltonian structures of the discrete integrable systems associated with the Lie algebra possessing degenerate Killing forms. Especially, it can be used to work out the Hamiltonian structures of some kinds of discrete integrable couplings. Then a kind of integrable coupling of the Toda hierarchy is obtained and its Hamiltonian structure is worked out by using the discrete quadratic-form identity. Moreover, the Liouville integrability of the integrable coupling is demonstrated.