Two-Photon Algebra and Integrable Hamiltonian Systems

In a recent paper [1], a systematic construction of integrable Hamiltonians with coalgebra symmetry has been proposed. Such procedure can be applied to any Poisson coalgebra (A,∆) with generators Xi, i = 1, . . . , l and Casimir element C(X1, . . . , Xl) as follows. Let us consider the N -th coproduct ∆(Xi) of the generators and the m-th order (2 ≤ m ≤ N) coproducts ∆(m)(C) of the Casimir operator of the coalgebra (recall that the m-th coproduct is an algebra homomorphism that maps ∆(m) : A → A⊗A⊗ · · ·m) ⊗A). By making use of the structural properties of the coproduct it can be proven that { ∆(C),∆(Xi) } = 0, i = 1, . . . , l. (1.1)