Vertex operators for quantum groups and application to integrable systems

Abstract

Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{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), we recover by this construction the nonlinear Schrodinger equation and its Y(N) symmetry.