(1+1) Schrödinger Lie bialgebras and their Poisson-Lie groups

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrödinger algebra are explicitly derived and proved to be of coboundary type. Therefore, since all of them come from a classical r -matrix, the complete family of Schrödinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl (2) Lie bialgebras within the Schrödinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrödinger algebra, including their corresponding quantum universal R -matrices, are constructed.