  All Lie bialgebra structures for the (1+1)-dimensional centrally extended
Schrodinger algebra are explicitly derived and proved to be of the coboundary
type. Therefore, since all of them come from a classical r-matrix, the complete
family of Schrodinger 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 Schrodinger classification are
studied. As an application, new quantum (Hopf algebra) deformations of the
Schrodinger algebra, including their corresponding quantum universal
R-matrices, are constructed.
