Vertex operators for quantum groups and application to integrable systems

Starting with any R-matrix with spectral parameters, and obeying the Yang–Baxter equation and a unitarity condition, we construct the corresponding infinite-dimensional quantum group R in terms of a deformed oscillator algebra R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated with R, we show that R provides its integrals of motion. The construction can be applied to any infinite-dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N), the Yangian based on gl(N), using this construction we recover the nonlinear Schrodinger equation and its Y(N) symmetry.