New integrable differential-difference and fractional nonlinear dynamical systems and their algebro-analytical properties

A new metrized pseudo-difference operator algebra on the line is constructed that allows application of the Lie-algebraic Adler-Kostant-Symes approach to generate infinite hierarchies of integrable nonlinear differential-difference Hamiltonian systems. It is shown that the metrized pseudo-difference operator algebra has a metrized fractional generalization, which can be used to construct new nonlinear fractional differential-difference hierarchies of integrable Hamiltonian systems of Korteweg-de Vries, Nonlinear-Schrödinger and Kadomtsev-Petviashvili types.